H-infinity controller design using control object models

ABSTRACT

A design device which designs a controller in accordance with H infinity (H ∞) logic, in which formulae of generalized plants and formulae of control object models which are parts of the generalized plants are stored in a memory unit  3 . In a parameter calculating unit including comprising a frequency response calculation unit  4  and a scaling matrix calculation unit  5 , the frequency response calculation unit  4  calculates the frequency responses of the control object models, and the scaling matrix calculation unit  5  calculates a scaling matrix T in accordance with the frequency responses of the control object models so that the respective gains of the control object models are consistent. A controller calculation unit  6  calculates the parameters of a controller by applying the scaling matrix T to the generalized plants.

TECHNICAL FIELD

The present invention relates to a design device which designs amultivariable controller in accordance with H infinity (H ∞) logic.

BACKGROUND ART

In recent years, in the feedback control field, the H ∞ logic has oftenbeen utilized which allows the design of a controller in view of anerror between an actual control object and a numerical model of thecontrol object. In the conventional control logic, when designing acontrol system, a control object model that is represented by thetransfer function and the state equation is prepared, and the controlsystem is designed to stabilize the model. At this time, when there isan insignificant error between the actual control object and the model,a controller designed to stabilize the model can also stabilize theactual control object. However, when there is a significant errorbetween the actual control object and the model for some reason, thecontroller may not stabilize the actual control object.

In the H ∞ logic, even if there is an error between the actual controlobject and the numerical model for use with the design, when theinformation as to the error can be obtained, the controller forstabilizing the actual control object can be designed in view of theerror. It is said that the H ∞ logic is more likely to give the controlspecification intuitively in designing the control system, as comparedwith the conventional control logic. For example, in a case of designinga control system using the conventional control logic, its designspecification involved a pole in the closed loop system or a weightmatrix of evaluation function. However, the physical meanings of thesevalues were unclear, and it required a lot of trial and error to makethe settings.

On the contrary, in the H ∞ logic, the control specification can bedefined in accordance with the frequency responses of the closed loopsystem consisting of the control object and the controller. The H ∞logic has such an advantage, but is theoretically difficult, and has notbeen put into practical use in the current situation for the reasons ofrequiring the considerable knowledge to construct the actual controlsystem, and it being difficult to give the control specification to theobjects less treatable in accordance with the frequency responses in theprocess control and so forth.

DISCLOSURE OF THE INVENTION

A control object model used in designing a multivariable control systemhas a variety of magnitudes of error components ranging from themanipulated variables to controlled variables. In this way, there arevarious error components of the model to the manipulated variables. Inthe H ∞ logic, since the controller is designed with reference to thecomponent with a large gain of error, the component with a small gain oferror is prone to have a very conservative response, or to beexcessively stable. In some cases, it may be required to adjust theweight of control for each controlled variable to avoid the interferencebetween controlled variables. Thus, it is proposed to introduce amanipulated variable weight called a scaling matrix T in order to makethe error magnitudes of the control object models consistent and weightthe control for the controlled variables. However, in the conventionaldesign method, no general solution was not established to determine thescaling matrix T, resulting in the problem that the scaling matrix T wasdifficult to suitably choose. Since it was difficult to determine thescaling matrix T, there was another problem that the H ∞ logic wasdifficult to utilize in the design of the multivariable controller.

Also, in the H ∞ logic, it is required to determine a frequency weightcalled a sensitivity weight W_(s) to determine the set value followupcharacteristic of the closed loop system. However, since the H ∞ logicis a design method in the frequency domain, there was the problem thatthe design is easy in the control of the mechanical system, but is hardin the control system untreatable in the frequency domain such as theprocess control, whereby it is difficult to suitably select thesensitivity weight W_(s). Also, since it is difficult to give thecontrol specification in the frequency domain, and hard to determine thesensitivity weight W_(s), there was the problem that the H ∞ logic isdifficult to utilize in the design of the controller used in the processcontrol field.

As described above, there was conventionally the problem that the H ∞logic was hard to utilize in the design of the controller.

The present invention has been made in order to solve these problems,and has as its object to provide a design device that can easily designa controller in accordance with the H ∞ logic.

A design device of a controller according to the present inventioncomprises storage means for storing generalized plants, parametercalculating means for calculating the parameters of the components ofthe generalized plants in accordance with the response characteristic ofa control object model or the response characteristic of a closed loopsystem consisting of the control object model and the controller, andcontroller calculation means for calculating the parameters of thecontroller by applying the parameters to the generalized plants storedin the storage means.

In the design device of the controller according to one arrangement ofthe invention, the generalized plants have the control object models,and manipulated variable weight adjusting means for adjusting the inputof manipulated variables into the control object models, which isprovided in the former stage of the control object models, the parametercalculating means comprises frequency response calculation means forcalculating the frequency responses of the control object models, andscaling matrix calculation means for calculating a scaling matrix T fordetermining the weighting of the manipulated variables with themanipulated variable weight adjusting means in accordance with thefrequency responses of the control object models so that the respectivegains of the control object models are consistent, and the controllercalculation means calculates the parameters of the controller byapplying the scaling matrix T to the manipulated variable weightadjusting means of the generalized plants stored in the storage means.

Also, in the design device of the controller according to onearrangement of the invention, the generalized plants have a firstcontrol object model for the manipulated variables, a second controlobject model for the disturbance, and manipulated variable weightadjusting means for adjusting the input of manipulated variables intothe first control object model, which is provided in the former stage ofthe first control object model, the parameter calculating meanscomprises frequency response calculation means for calculating thefrequency responses of the first control object model and the secondcontrol object model, and scaling matrix calculation means forcalculating a scaling matrix T for determining the weighting of themanipulated variables with the manipulated variable weight adjustingmeans in accordance with the frequency responses of the first and secondcontrol object models so that the respective gains of the first controlobject model are consistent with the maximum value of the gains of thesecond control object model, and the controller calculation meanscalculates the parameters of the controller by applying the scalingmatrix T to the manipulated variable weight adjusting means of thegeneralized plants stored in the storage means.

Also, in the design device of the controller according to onearrangement of the invention, the generalized plants stored in thestorage means have control variable weight adjusting means for adjustingthe controlled variables inside a closed loop system consisting of themanipulated variable weight adjusting means, the control object modelfor the manipulated variables and the controller, and the design devicehas setting means for setting a weight matrix S for determining theweighting of the controlled variables with the control variable weightadjusting means.

Also, in the design device of the controller according to onearrangement of the invention, the generalized plants stored in thestorage means have control variable weight adjusting means for adjustingthe controlled variables in the former or latter stage of frequencysensitivity weight adjusting means to determine the set value followupcharacteristic of a closed loop system consisting of the manipulatedvariable weight adjusting means, the control object models for themanipulated variables and the controller, and the design device hassetting means for setting a weight matrix S for determining theweighting of the controlled variables with the control variable weightadjusting means.

In the design device of the controller according to one arrangement ofthe invention, the parameter calculating means has setting means forsetting the transient response characteristic of the closed loop systemand frequency sensitivity weight calculation means for calculating thefrequency sensitivity weight to determine the set value followupcharacteristic of the closed loop system in accordance with thetransient response characteristic of the closed loop system, and thecontroller calculation means calculates the parameters of the controllerby applying the frequency sensitivity weight to the generalized plantsstored in the storage means.

In the design device of the controller according to one arrangement ofthe invention, the frequency sensitivity weight calculation meanscalculates the frequency sensitivity weight in accordance with thetransient response characteristic of the closed loop system, and adesign index that the H ∞ norm of a transfer function of the closed loopsystem from the set value to the deviation, multiplied by the frequencysensitivity weight, is less than 1.

In the design device of the controller according to one arrangement ofthe invention, the setting means approximates the transient responsecharacteristic of the closed loop system with a first-order lagcharacteristic.

In the design device of the controller according to one arrangement ofthe invention, the setting means approximates the transient responsecharacteristic of the closed loop system with a second-order systemcharacteristic.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram showing the configuration of a design deviceof a controller in a first embodiment of the present invention;

FIG. 2 is a block diagram showing the configuration of a model of anactual control object as represented in the formulae;

FIG. 3 is a block diagram showing the configuration of the conventionalgeneralized plants;

FIG. 4 is a block diagram showing the configuration of the generalizedplants for use with the design device of the invention;

FIG. 5 is a block diagram showing the configuration of a robust controlsystem with a controller added to the generalized plants of FIG. 4;

FIG. 6 is a block diagram showing the configuration of an actualcontroller containing the controller designed using the design device inthe first embodiment of the invention;

FIG. 7 is a block diagram showing an additive error in a numerical modelof the control object;

FIG. 8 is a block diagram showing the configuration of a closed loopsystem used in determining the sensitivity weight;

FIG. 9 is a graph for explaining the operation of a scaling matrix inthe first embodiment of the invention;

FIG. 10 is a graph for explaining the operation of a scaling matrix in asecond embodiment of the invention;

FIG. 11 is a block diagram showing the configuration of a design deviceof a controller in a third embodiment of the present invention;

FIG. 12 is a block diagram showing the configuration of the generalizedplants in the third embodiment of the present invention;

FIG. 13 is a block diagram showing the configuration of an actualcontroller containing the controller designed using the design device inthe third embodiment of the invention;

FIG. 14 is a block diagram showing the configuration of the generalizedplants in a fourth embodiment of the present invention;

FIG. 15 is a block diagram showing the configuration of a design deviceof a controller in a fifth embodiment of the present invention;

FIG. 16 is a block diagram showing the configuration of a model of anactual control object as represented in the formulae;

FIG. 17 is a block diagram showing the configuration of the conventionalgeneralized plants;

FIG. 18 is a block diagram showing the configuration of the generalizedplants for use with the design device of the invention;

FIG. 19 is a block diagram showing the configuration of a robust controlsystem with a controller added to the generalized plants of FIG. 18;

FIG. 20 is a block diagram showing the configuration of an actualcontroller containing the controller designed using the design device ofthe invention;

FIG. 21 is a block showing an additive error in a numerical model of thecontrol object;

FIG. 22 is a block diagram showing the configuration of a closed loopsystem used in determining the sensitivity weight; and

FIG. 23 is a graph showing the time response characteristic when theclosed loop system is approximated by a second-order system.

BEST MODE OF CARRYING OUT THE INVENTION FIRST EMBODIMENT

The present invention will be described below in detail by way ofembodiment with reference to the accompanying drawings. FIG. 1 is ablock diagram showing the configuration of a design device of acontroller in a first embodiment of the invention. The design device ofFIG. 1 comprises a control object model input unit 1 for inputting theparameters of a control object model, a control object modelregistration unit 2 for registering the model parameters in a memoryunit described later, the memory unit 3 for storing formulae ofgeneralized plants and formulae of control object models which are partsof the generalized plants, a frequency response calculation unit 4 forcalculating the frequency responses of the control object models, ascaling matrix calculation unit 5 for calculating a scaling matrix T formaking consistent the magnitudes of errors of the control object modelsso that the respective gains of the control object models are consistentwith the maximum value of the gains of the control object models, and acontroller calculation unit 6 for calculating the parameters of acontroller by applying the scaling matrix T to the generalized plantsstored in the memory unit 3.

An algorithm for designing the controller in accordance with the H ∞logic is effected based on the generalized plants as represented usingthe control objects. Therefore, the generalized plants will be firstdescribed. FIG. 2 is a block diagram showing the configuration of amodel of an actual control object as represented in the formulae. Anumerical model of the control object as shown in FIG. 2 is composed ofa first control object model 11 for the manipulated variable u and asecond control object model 12 for the disturbance w. Reference sign Pudenotes a transfer function of the model 11 and reference sign Pwdenotes a transfer function of the model 12. The models 11, 12 areobtained as a result of the model identification using the data obtainedby a step response test for the actual control object. The controlledvariable y that is the output of control object is a sum of the outputsfrom the models 11 and 12.

FIG. 3 shows the configuration of the conventional generalized plantscontaining such numerical model of control object. The generalizedplants involve providing a frequency weight called a sensitivity weightW_(s) used to determine the set value followup characteristic and afrequency weight called a complementary sensitivity weight W_(t) used todetermine the robust stability, and introducing the set value r, and theoutputs z1, z2 in addition to the input (manipulated variable) u ofcontrol object, input (disturbance) w, and output (controlled variable)y, to realize both the set value followup characteristic and the robuststability as shown in FIG. 3. Deviation e (=y−r) is an observationquantity or the input into the controller (not shown). Reference numeral13 denotes a block (frequency sensitivity weight adjusting means)representing the sensitivity weight W_(s), and reference sign Z₁ denotesan output for effecting the evaluation of the set value followupcharacteristic. Also, reference numeral 14 denotes a block representingthe complementary sensitivity weight W_(t), and reference sign Z₂denotes an output for effecting the evaluation of the robust stability.

Conventionally, in the generalized plants as shown in FIG. 3, thecomplementary sensitivity weight W_(t) is determined by estimating theuncertainty of the models on the basis of the numerical models ofcontrol objects, and the sensitivity weight W_(s) is determined bydirectly specifying the frequency characteristic in view of the followupability to the set value r, whereby the parameters of the controller aredetermined through the γ iteration. However, since employing thegeneralized plant of FIG. 3, the controller is designed on the basis ofa larger gain due to differences in the gains for the plant outputs ofthe manipulated variables, the obtained controller is likely to be veryconservative, or excessively stable. Also, since the set value followupcharacteristic and the disturbance response characteristic are usuallyreciprocal, it is preferred to design with the weight according to thepurpose rather than with the same weight. Further, in the generalizedplants of FIG. 3 containing no integral element, some steady-statedeviation arises. If the sensitivity weight W_(s) is provided with theintegral characteristic, the controller can have the integralcharacteristic, but the generalized plants become unstable, not leadingto the normal H ∞ problem.

Therefore, in this embodiment, the generalized plants as shown in FIG. 4are considered. In FIG. 4, reference sign M denotes a scaling matrix foradjusting the influence of disturbance w on the controlled variable y,reference sign T denotes a scaling matrix for making the magnitude oferror of the control object model consistent, and reference sign α⁻¹Idenotes a weight for providing the controller with the integralcharacteristic to eliminate the steady-state deviation. Herein, α(s) isdefined as α(s)=s/(s+a). Where s is a Laplace operator, and a (>0) isany real number. Reference numeral 15 denotes a block representing thescaling matrix M, reference numeral 16 denotes a block (manipulatedvariable weight adjusting means) representing the scaling matrix T, andreference numeral 17 denotes a block representing the weight α⁻¹I.Deviation e₂ is deviation e multiplied by the weight α⁻¹I, and inputinto the controller. FIG. 5 shows the configuration of a robust controlsystem in which a controller K is added to the above generalized plant.In FIG. 5, reference numeral 18 denotes a block representing thecontroller K.

The design device of the controller in this embodiment is aimed atdetermining the parameters of the controller K such that the controlledvariable y that is the output of control object follows the set value r,the influence of disturbance w is removed, and the control object isstabilized even if it is fluctuated or there is an error in the model ofcontrol object. The H ∞ problem can be regarded as the problem ofreducing the H ∞ norm (gain) of the transfer function from (r, w) to(z₁, z₂). That is, the set value followup characteristic, the robuststability and the disturbance suppression may be considered in thefollowing way.

(A) Set value followup characteristic: If the H ∞ norm (gain) of thetransfer function from the set value r to deviation e (more correctly,transfer function with the set value r multiplied by the frequencyweight α⁻¹W_(s) from r to z₁) is reduced, the deviation e can bedecreased, so that the set value followup characteristic can be madebetter. Herein, α⁻¹W_(s) is the frequency weight for restricting thefollowup band (e.g., followup only in the low band).

(B) Robust stability: There is an error between the actual controlobject and its model due to characteristic variations of the controlobject or the error at the time of modeling. The maximum value of theerror from the identified model is estimated as Δ(s), and the controllerK is designed such that the H ∞ norm from the set value r to z₂ issmaller than or equal to 1, employing the complementary sensitivityweight W_(t)(s) such as |Δ(jω)|<|W_(t)(jω) for this error, whereby therobust stabilization can be achieved.

(C) Disturbance suppression: If the H ∞ norm (gain) of the transferfunction from disturbance w to deviation e (more correctly, transferfunction with the disturbance w multiplied by the frequency weightα⁻¹W_(s) from w to z₁) is reduced, the deviation e can be decreased evenif the disturbance w enters, whereby the disturbance suppression can beameliorated.

Next, it is supposed that the state space representation of thegeneralized plant as shown in FIG. 4 is given by:{dot over (x)} _(p) =A _(p) x _(p) +B _(p1) Mw+B _(p2) Tu  (1)y=C _(p) x _(p) +D _(p1) Mw+D _(p2) Tu  (2)

In the above expressions (1) and (2), x_(p) is the quantity of state,and A_(p), B_(p1), B_(p2), C_(p), D_(p1) and D_(p2) are the parametersof the numerical models 11, 12 of control objects. From the expression(2), the deviation e can be obtained in the following expression.e=y−r=C _(p) x _(p) +D _(p1) Mw+D _(p2) Tu−r  (3)

With the configuration of the generalized plant as shown in FIG. 4, theoutputs z₁′ and z₂′ can be defined in the following expressions.z₁′=e₂  (4)z₂′=u  (5)

The frequency weight for providing the controller K with the integralcharacteristic can be defined in the following expression, using theexpression (3).

$\begin{matrix}\begin{matrix}{{\overset{.}{x}}_{\alpha} = {{A_{\alpha}x_{\alpha}} + {B_{\alpha}e}}} \\{= {{A_{\alpha}x_{\alpha}} + {B_{\alpha}C_{p}x_{p}} + {B_{\alpha}D_{p1}{Mw}} + {B_{\alpha}D_{p2}{Tu}} - {B_{\alpha}r}}}\end{matrix} & (6)\end{matrix}$

$\begin{matrix}\begin{matrix}{e_{2} = {{C_{\alpha}x_{\alpha}} + {D_{\alpha}e}}} \\{= {{C_{\alpha}x_{\alpha}} + {D_{\alpha}C_{p}x_{p}} + {D_{\alpha}D_{p1}{Mw}} + {D_{\alpha}D_{p2}{Tu}} - {D_{\alpha}r}}}\end{matrix} & (7)\end{matrix}$

In the expressions (6) and (7), x_(α) is the quantity of state of α⁻¹I,and A_(α), B_(α), C_(α) and D_(α) are the parameters of α⁻¹I. Arrangingthe above expressions and representing them in a state space, thefollowing three expressions can be obtained.

$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}t}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} = {{\begin{bmatrix}A_{p} & 0 \\{B_{\alpha}C_{p}} & A_{\alpha}\end{bmatrix}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} + {\left\lbrack {\begin{matrix}0 \\{- B_{\alpha}}\end{matrix}\begin{matrix}{B_{p1}M} \\{B_{\alpha}D_{p1}M}\end{matrix}\begin{matrix}{B_{p2}T} \\{B_{\alpha}D_{p2}T}\end{matrix}} \right\rbrack\begin{bmatrix}r \\w \\u\end{bmatrix}}}} & (8)\end{matrix}$

$\begin{matrix}{\begin{bmatrix}{z_{1}}^{\prime} \\{z_{2}}^{\prime}\end{bmatrix} = {{\begin{bmatrix}{D_{\alpha}C_{p}} & C_{\alpha} \\0 & 0\end{bmatrix}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} + {\left\lbrack {\begin{matrix}{- D_{\alpha}} & {D_{\alpha}D_{p1}M} \\0 & 0\end{matrix}\begin{matrix}{D_{\alpha}D_{p2}T} \\I\end{matrix}} \right\rbrack\begin{bmatrix}r \\w \\u\end{bmatrix}}}} & (9)\end{matrix}$

$\begin{matrix}{e_{2} = {{\begin{bmatrix}{D_{\alpha}C_{p}} & C_{\alpha}\end{bmatrix}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} + {\begin{bmatrix}{- D_{\alpha}} & {D_{\alpha}D_{p1}M} & {D_{\alpha}D_{p2}T}\end{bmatrix}\begin{bmatrix}r \\w \\u\end{bmatrix}}}} & (10)\end{matrix}$

Representing the expressions (8), (9) and (10) in Doile's notation, thefollowing expressions can be obtained.

$\begin{matrix}{{G(s)} = \begin{bmatrix}A & B_{1} & B_{2} \\C_{1} & D_{11} & D_{12} \\C_{2} & D_{21} & D_{22}\end{bmatrix}} & (11)\end{matrix}$

Parameter A can be represented as follows:

$\begin{matrix}{A = \begin{bmatrix}A_{p} & 0 \\{B_{\alpha}C_{p}} & A_{\alpha}\end{bmatrix}} & (12)\end{matrix}$

Parameter B₁ and B₂ can be represented as follows:

$\begin{matrix}{{B_{1} = \begin{bmatrix}0 & {B_{p1}M} \\{- B_{\alpha}} & {B_{\alpha}D_{p1}M}\end{bmatrix}},{B_{2} = \begin{bmatrix}{B_{p2}T} \\{B_{\alpha}D_{p2}T}\end{bmatrix}}} & (13)\end{matrix}$

Also, parameters C₁ and C₂ can be represented as follows:

$\begin{matrix}{{C_{1} = \begin{bmatrix}{D_{\alpha}C_{p}} & C_{\alpha} \\0 & 0\end{bmatrix}},{C_{2} = \left\lfloor \begin{matrix}{D_{\alpha}C_{p}} & C_{\alpha}\end{matrix} \right\rfloor}} & (14)\end{matrix}$

And the parameters D₁₁, D₁₂, D₂₁, D₂₂ can be represented as follows:

$\begin{matrix}{{{D_{11} = \begin{bmatrix}{- D_{\alpha}} & {D_{\alpha}D_{p1}M} \\0 & 0\end{bmatrix}},{D_{12} = \begin{bmatrix}{D_{\alpha}D_{p2}T} \\I\end{bmatrix}}}{{D_{21} = \left\lfloor \begin{matrix}{- D_{\alpha}} & {D_{\alpha}D_{p1}M}\end{matrix} \right\rfloor},{D_{22} = {D_{\alpha}D_{p2}T}}}} & (15)\end{matrix}$

The sensitivity weight W_(s) and the complementary sensitivity weightW_(t) are designed, and multiplied by the output parts of the expression(11). Through γ iteration, the controller K is obtained in the statespace representation. Herein, the output parts of the expression (11)signify the parts corresponding to the outputs z₁′, z₂′ in FIG. 4.Hence, an output equation of the parameters C₁, D₁₁, D₁₂ of theexpression (11) may be multiplied by a diagonal matrix Q as representedin the following expression having the diagonal elements of sensitivityweight W_(s) and complementary sensitivity weight W_(t) from the leftside. Thus, the parameters of the controller K can be calculated.

$\begin{matrix}{Q = \begin{bmatrix}W_{s} & 0 \\0 & W_{t}\end{bmatrix}} & (16)\end{matrix}$

The controller K is a solution of the H ∞ control problem with thegeneralized plants, and the actual controller mounted on the plants suchas a distillation tower is the controller K multiplied by weight α⁻¹Iand scaling matrix T, as shown in FIG. 6.

Next, a method for determining the complementary sensitivity weightW_(t) in this embodiment will be described below. The control object isvaried in the characteristics, depending on the driving conditions.Normally, the control design is made based on a certain model, but inthe robust control design, the variation of control object and themagnitude of error of modeling are contained beforehand in the controldesign, and the controller is designed to be stable without muchdeterioration in the control performance even if there is any variationor error. FIG. 7 shows an additive error for the model 11 of controlobject. In FIG. 7, reference numeral 19 denotes a block representing theadditive error Δ. In the robust control design, a variation in thecharacteristic of the control object owing to the driving conditions anda model error due to lower dimension of the model 11 are represented asthe additive error Δ as shown in FIG. 7. If the characteristics of thecontrol object are deviated from the model 11 due to this additive errorΔ, the controller is designed such that the controller output may bestable. To this end, the complementary sensitivity weight W_(t) may bedetermined to cover the additive error Δ. The general expression of thiscomplementary sensitivity weight W_(t) is shown in the followingexpression. Since the change of the model 12 is not related with thestability of the system, it is supposed that the model 11 alone isvaried in designing the controller.

$\begin{matrix}{W_{t} = \begin{bmatrix}W_{t1} & 0 & 0 & \cdots & 0 \\0 & W_{t2} & 0 & \cdots & 0 \\0 & 0 & W_{t3} & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \vdots \\0 & 0 & 0 & \cdots & W_{tN}\end{bmatrix}} & (17)\end{matrix}$

In this embodiment, for the additive error Δ with scaled magnitude,employing the scaling matrix T, the element of the complementarysensitivity weight W_(t) is the maximum value Gmax of the gains of errorΔ multiplied by a safety factor δ (δ is equal to 1, for example).Namely, the elements (weights) W_(t1), W_(t2), W_(t3), . . . , W_(tN)are defined as follows.W _(t1) =W _(t2) =W _(t3) =W _(tN)=(1+δ)G max  (18)

This embodiment is involved with the multivariable control system, andassuming that the number of manipulated variables u is N (N is apositive integer), the complementary sensitivity weight W_(t) is N×Nmatrix. Where W_(tN) is the weight for the N-th manipulated variableu_(N).

Next, a method for determining the sensitivity weight W_(s) in thisembodiment will be described below. First of all, a general expressionof the sensitivity weight W_(s) is shown in the following expression.

$\begin{matrix}{W_{s} = \begin{bmatrix}W_{s1} & 0 & 0 & \cdots & 0 \\0 & W_{s2} & 0 & \cdots & 0 \\0 & 0 & W_{s3} & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \vdots \\0 & 0 & 0 & \cdots & W_{sL}\end{bmatrix}} & (19)\end{matrix}$

Assuming that the number of controlled variables y is L (L is a positiveinteger), the sensitivity weight W_(s) is L×L matrix. Element W_(sL) ofthe sensitivity weight W_(s) is the weight for the L-th controlledvariable y_(L). In order to determine the sensitivity weight W_(s), aclosed loop system of FIG. 8 in which the robust control system of FIG.5 is simplified is considered. In FIG. 8, reference numeral 11 a denotesa block representing a numerical model P of control object, andreference numeral 13 a denotes a block representing the frequency weightW_(s)′. Supposing that the sensitivity function indicating the controlperformance mainly regarding the quick-response property such as the setvalue followup ability or disturbance suppression is S(s), the smallergain |S(jω)| of the sensitivity function S(s) is preferable because themodel variations have less effect on the set value response. If thecontrol specification at each frequency is given by S_(spec)(ω), thefollowing condition concerning the sensitivity function S(s) can beobtained.|S(jω)|<S _(spec)(ω); ∀ω  (20)

Where ∀ω means that the expression (20) holds for all frequencies ω.Employing this sensitivity function S(s), the design index of thecontroller K in view of the set value followup characteristic is asfollows:∥W _(sL)′(s)S(s)∞<1  (21)

The frequency weight W_(sL)′ (s) is W_(sL) (s) multiplied by α⁻¹ (s),and is defined as follows:W _(sL)′(s)=α⁻¹(s)W _(sL)(s)  (22)

The expression (21) indicates that the H ∞ norm of the transfer functionof the closed loop system of FIG. 8 from the set value r to thedeviation e (more correctly, the transfer function with the set value rmultiplied by the frequency weight α⁻¹(s) W_(sL) (s) from r to z₁) isless than 1. By setting the weight W_(sL)(s) to satisfy this expression(21), the controller K can be designed in view of the set value followupcharacteristic.

Next, a method for determining the scaling matrix M will be describedbelow. A general expression of the scaling matrix M is shown in thefollowing expression.

$\begin{matrix}{M = \begin{bmatrix}M_{1} & 0 & 0 & \cdots & 0 \\0 & M_{2} & 0 & \cdots & 0 \\0 & 0 & M_{3} & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \vdots \\0 & 0 & 0 & \cdots & M_{J}\end{bmatrix}} & (23)\end{matrix}$

Assuming that the number of disturbance w is J (J is a positiveinteger), the scaling matrix M is J×J matrix. Element M_(J) of thescaling matrix M is the weight for the J-th disturbance w_(J), with theinitial value being 1. Each element M_(J) is an adjustment parameter fordetermining the disturbance suppression performance by adjusting theinfluence of each disturbance w_(J) on the controlled variable y. Thatis, when the suppression of the particular disturbance w is desired tobe intensified, the element M_(J) concerning this disturbance w is madelarger than 1.

Next, a method for determining the scaling matrix T in this embodimentwill be described below. A general expression of the scaling matrix T isshown in the following expression.

$\begin{matrix}{T = \begin{bmatrix}T_{1} & 0 & 0 & \cdots & 0 \\0 & T_{2} & 0 & \cdots & 0 \\0 & 0 & T_{3} & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \vdots \\0 & 0 & 0 & \cdots & T_{N}\end{bmatrix}} & (24)\end{matrix}$

Assuming that the number of manipulated variables u is N (N is apositive integer), the scaling matrix T is N×N matrix. Element T_(N) ofthe scaling matrix T is the weight for the N-th manipulated variableu_(N). Each element T_(N) is determined such that the magnitudes of thegains of the control object model 11 are as equal as possible. Morespecifically, each element T_(N) is determined as follows:

$\begin{matrix}{T_{N} = {{\frac{1}{L} \cdot \frac{\max\left( {{{G_{y1u1}}}_{\infty},{{G_{y1u2}}}_{\infty},\ldots\mspace{14mu},{{G_{y1uN}}}_{\infty}} \right)}{{{G_{y\; 1u\; N}}}_{\infty}}} + {\frac{1}{L} \cdot \frac{\max\left( {{{G_{y2u1}}}_{\infty},{{G_{y2u2}}}_{\infty},\ldots\mspace{14mu},{{G_{y2uN}}}_{\infty}} \right)}{{{G_{y\; 2{uN}}}}_{\infty}}} + \ldots + {\frac{1}{L} \cdot \frac{\max\left( {{{G_{yLu1}}}_{\infty},{{G_{yLu2}}}_{\infty},\ldots\mspace{14mu},{{G_{yLuN}}}_{\infty}} \right)}{{{G_{yLuN}}}_{\infty}}}}} & (25)\end{matrix}$

In the expression (25), G_(yLuN) is the transfer function of the controlobject model 11 of FIG. 4 from the N-th manipulated variable u_(N) tothe L-th controlled variable y_(L), and ∥G_(yLuN)∥∞ is the H ∞ norm(gain) of the same transfer function. max (∥G_(yLu1)∥∞, ∥G_(yLu2)∥∞, . .. , ∥G_(yLuN)∥∞) means to select the maximum value from among the H ∞norms ∥G_(yLu1)∥∞, ∥G_(yLu2)∥∞, . . . , and ∥G_(yLuN)∥∞. To obtain the H∞ norm ∥G_(yLuN)∥∞, the model 11 as presented in the state equationrepresentation may be transformed into the transfer functionrepresentation given by the following expression, and the gain of eachfrequency may be calculated from this transfer function.

$\begin{matrix}{\begin{bmatrix}y_{1} \\y_{2} \\y_{3} \\\vdots \\y_{L}\end{bmatrix} = {\begin{bmatrix}G_{y1u1} & G_{y1u2} & G_{y1u3} & \cdots & G_{y1uN} \\G_{y2u1} & G_{y2u2} & G_{y2u3} & \cdots & G_{y2uN} \\G_{y3u1} & G_{y3u2} & G_{y3u3} & \cdots & G_{y3uN} \\\vdots & \vdots & \vdots & ⋰ & \vdots \\G_{yLu1} & G_{yLu2} & G_{yLu3} & \cdots & G_{yLuN}\end{bmatrix}\begin{bmatrix}u_{1} \\u_{2} \\u_{3} \\\vdots \\u_{N}\end{bmatrix}}} & (26)\end{matrix}$

Thereby, the H ∞ norm ∥G_(yLuN)∥∞ can be obtained for each of themanipulated variables u and the controlled variables y, and the elementT_(N) of the scaling matrix T can be obtained from the expression (25).

Referring now to FIG. 9, the operation of the scaling matrix T will bedescribed below. FIG. 9A shows the gain characteristic of the controlobject model 11 (frequency response characteristic of the model 11). InFIG. 9, three sorts of gain characteristic are only shown to simplifythe description, but if the number of manipulated variables u is N andthe number of controlled variables y is L, N×L sorts of gains exist. Asshown in FIG. 9A, in the case where there is no scaling matrix T, itwill be found that the gains of the control object model 11 areinconsistent. In general, if the gains of the control object model areinconsistent, the magnitudes of the errors of the control object modelare correspondingly inconsistent. Since the complementary sensitivityweight W_(t) is determined to cover the additive error Δ, as previouslydescribed, it follows that the controller is designed on the basis ofthe model with larger error, whereby the controller obtained is likelyto be very conservative, or excessively stable.

Thus, the magnitudes of the gains are made consistent, employing thescaling matrix T. FIG. 9B shows the gain characteristic of the controlobject model 11 in the case where the scaling matrix T of thisembodiment is provided. ∥G_(yumax)∥∞ is the maximum value of the gainsof the model 11. As will be clear from FIG. 9B, the method fordetermining the scaling matrix T of this embodiment as shown in theexpressions (24) and (25) involves determining the scaling matrix T suchthat the gains are consistent with the gain maximum value ∥G_(yumax)∥∞(more correctly, the approximation of the gain maximum value) of themodel 11.

Referring now to FIG. 1, the above operation will be described below.The parameters of the control object model 11 are set in the controlobject model input unit 1 by the user of the design device. The controlobject model registration unit 2 registers the parameters input from thecontrol object model input unit 1 into the formulae of the controlobject model stored beforehand in the memory unit 3. The control objectmodel input unit 1 and the control object model registration unit 2constitute model setting means for setting the control object model. Thememory unit 3 stores the formulae of the generalized plants of FIG. 4and the formulae of the control object models that are parts of thegeneralized plants, as described in the expressions (1) to (15). Thefrequency response calculation unit 4 transforms the model 11 aspresented in the state equation representation which is registered inthe memory unit 3 into the transfer function representation, andcalculates the gain at each frequency from the transfer function.Subsequently, the scaling matrix calculation unit 5 calculates thescaling matrix T, employing the expressions (24) and (25), on the basisof the gains calculated in the frequency response calculation unit 4,and outputs it to the controller calculation unit 6. The controllercalculation unit 6 registers the scaling matrix T in the formulae of thegeneralized plants stored in the memory unit 3, and through γ iteration,calculates the parameters of the controller K. At this time, thecomplementary sensitivity weight W_(t), the sensitivity weight W_(s) andthe scaling matrix M are preset in the generalized plants in the memoryunit 3. In this way, the controller K can be designed.

As previously described, the conventional method has no general way ofdetermining the scaling matrix T, and empirically determined the scalingmatrix T. On the contrary, in this embodiment, the scaling matrix T iscalculated, in accordance with the frequency responses of the controlobject models 11, such that the respective gains of the control objectmodels are consistent with the gain maximum value (more correctly, theapproximation of the gain maximum value) of the control object model 11.Hence, the scaling matrix T can be easily determined. In this way, themultivariable controller can be easily designed in accordance with the H∞ logic which is superior in the set value followup characteristic, andcapable of stabilizing the controller even when the control objects arevaried or the control object models 11 have some error. Consequently,the multivariable control system can be easily designed in view of thevariations of the control objects and the uncertainty of the numericalmodels. Also, the multivariable controller can be realized utilizing thefeatures of the H ∞ control with smaller computational load duringexecution of the control, and with which the small-scale control systemcan be implemented.

In this embodiment, the scaling matrix T is determined such that therespective gains of the control object models are consistent with thegain maximum value (more correctly, the approximation of the maximumvalue) of the control object model 11, but the scaling matrix T may bedetermined such that the respective gains are consistent with the gainminimum value or gain average value of the control object model 11. Tomake the respective gains consistent with the gain minimum value (morecorrectly, the approximation of the minimum value), max in theexpression (25) may be replaced with min for selecting the minimum valueof ∥G_(yLu1)∥∞, ∥G_(yLu2)∥∞, . . . , and ∥G_(yLuN)∥∞. And to make therespective gains consistent with the gain average value (more correctly,the approximation of the average value), max in the expression (25) maybe replaced with E for acquiring the average value of ∥G_(yLu1)∥∞,∥G_(yLu2)∥∞, . . . , and ∥G_(yLuN)∥∞.

SECOND EMBODIMENT

In the first embodiment, the disturbance w is not considered, but thecontrol object model for the disturbance w may be obtained. Thus, inthis second embodiment, a method for determining the scaling matrix T inview of the influence of disturbance w in such a case will be describedbelow. In this second embodiment, the general expression of the scalingmatrix T can be given by the expression (24) in the same manner as inthe first embodiment.

And in this second embodiment, each element T_(N) of the scaling matrixT is determined in the following expression.

$\begin{matrix}\begin{matrix}{T_{N} = {{\frac{1}{L} \cdot \frac{\max\left( {{G_{y\; 1w\; 1}}_{\infty},{G_{y\; 1w\; 2}}_{\infty},\ldots,{G_{y\; 1w\; J}}_{\infty}} \right)}{{G_{y\; 1u\; N}}_{\infty}}} +}} \\{{\frac{1}{L} \cdot \frac{\max\left( {{G_{y\; 2w\; 1}}_{\infty},{G_{y\; 2w\; 2}}_{\infty},\ldots,{G_{y\; 2w\; J}}_{\infty}} \right)}{{G_{y\; 2u\; N}}_{\infty}}} + \cdots +} \\{\frac{1}{L} \cdot \frac{\max\left( {{G_{y\;{Lw}\; 1}}_{\infty},{G_{y\;{Lw}\; 2}}_{\infty},\ldots,{G_{y\;{Lw}\; J}}_{\infty}} \right)}{{G_{y\;{Lu}\; N}}_{\infty}}}\end{matrix} & (27)\end{matrix}$

In the expression (27), G_(yLwJ) is the transfer function of the controlobject model 12 of FIG. 4 from the J-th disturbance w_(j) to the L-thcontrolled variable y_(L), and ∥G_(yLwJ)∥∞ is the H ∞ norm (gain) of thesame transfer function. max (∥G_(yLw1)∥∞, ∥G_(yLw2)∥∞, . . . ,∥G_(yLwJ)∥∞) means to select the maximum value from the H ∞ norms∥G_(yLw1)∥∞, ∥G_(yLw2)∥∞, . . . , and ∥G_(yLwJ)∥∞. To obtain the H ∞norm ∥G_(yLwJ)∥∞, the model 12 as presented in the state equationrepresentation may be transformed into the transfer functionrepresentation as in the following expression, and the gain of eachfrequency may be calculated from this transfer function.

$\begin{matrix}{\begin{bmatrix}y_{w\; 1} \\y_{w\; 2} \\y_{w\; 3} \\\vdots \\y_{wL}\end{bmatrix} = {\begin{bmatrix}G_{y\; 1w\; 1} & G_{y\; 1w\; 2} & G_{y\; 1w\; 3} & \cdots & G_{y\; 1w\; J} \\G_{y\; 2w\; 1} & G_{y\; 2w\; 2} & G_{y\; 2w\; 3} & \cdots & G_{y\; 2w\; J} \\G_{y\; 3w\; 1} & G_{y\; 3w\; 2} & G_{y\; 3w\; 3} & \cdots & G_{y\; 3w\; J} \\\vdots & \vdots & \vdots & ⋰ & \vdots \\G_{y\;{Lw}\; 1} & G_{y\;{Lw}\; 2} & G_{y\;{Lw}\; 3} & \cdots & G_{y\;{Lw}\; J}\end{bmatrix}\begin{bmatrix}w_{1} \\w_{2} \\w_{3} \\\vdots \\w_{J}\end{bmatrix}}} & (28)\end{matrix}$

In the expression (28), y_(wL) is the output of the control object model12 for the disturbance w. Thereby, the H ∞ norm ∥G_(yLwJ)∥∞ can beobtained for each of the disturbance w and the controlled variables y,and the element T_(N) of the scaling matrix T can be obtained from theexpression (27).

Referring now to FIG. 10, the operation of the scaling matrix T will bedescribed below. FIG. 10A shows the gain characteristic of the controlobject model 12 (frequency response characteristic of the model 12). InFIG. 10A, three sorts of gain characteristic are only shown to simplifythe description, but if the number of disturbance w is J and the numberof controlled variables y is L, J×L sorts of gain exist. ∥G_(ywmax)∥∞ isthe maximum value of the gains of the model 12.

On the other hand, FIG. 10B shows the gain characteristic of the controlobject model 11. As shown in FIG. 10B, it will be found that the gainmaximum value of the control object model 12 and the gains of thecontrol object model 11 are inconsistent.

FIG. 10C shows the gain characteristic of the control object model 11 inthe case where the scaling matrix T of this embodiment is provided. Aswill be clear from FIG. 10C, the method for determining the scalingmatrix T of this embodiment as shown in the expressions (24) and (27)involves determining the scaling matrix T such that the respective gainsof the model 11 are consistent with the gain maximum value ∥G_(ywmax)∥∞(more correctly, the approximation of the gain maximum value) of themodel 12.

The scaling matrix T for the manipulated variable u is included in theclosed loop system in mounting the controller. Accordingly, there is ameaning in making the magnitudes of the gains of the model 11consistent, and it is not necessarily of importance with which the gainfrom the manipulated variable u to the controlled variable y isconsistent. The previous first embodiment illustrates one instance withwhich the respective gains of the control object models are consistent.On the contrary, in this second embodiment, because the input ofdisturbance is considered, it is necessary that the influence of theinput disturbance w is suppressed by the manipulated variable u from theviewpoint of suppressing the disturbance w. Thus, in this secondembodiment, to cope with the worst situation, the scaling matrix T isdetermined such that the gains of the model 11 are consistent with thegain maximum value ∥G_(ywmax)∥∞ (more correctly, the approximation ofthe gain maximum value) of the model 12.

In this second embodiment, the configuration of the design device isalmost equivalent to that of the first embodiment. Thus, the operationof the design device of this second embodiment will be described belowwith reference to FIG. 1.

The parameters of the control object model (model 11, 12 in this secondembodiment) are set in the control object model input unit 1 by the userof the design device. The control object model registration unit 2registers the parameters input from the control object model input unit1 into the formulae of the control object model stored beforehand in thememory unit 3. The frequency response calculation unit 3 transforms themodel 11, 12 as presented in the state equation representation which isregistered in the memory unit 3 into the transfer functionrepresentation, and calculates the gain at each frequency from thetransfer function. Subsequently, the scaling matrix calculation unit 5calculates the scaling matrix T, employing the expressions (24) and(27), on the basis of the gain calculated in the frequency responsecalculation unit 4, and outputs it to the controller calculation unit 6.The operation of the controller calculation unit 6 is exactly the sameas in the first embodiment. In this way, the controller K can bedesigned.

As described above, in this second embodiment, the scaling matrix T iscalculated, on the basis of the frequency response of the first andsecond control object models 11, 12, such that the respective gains ofthe first control object model 11 are consistent with the maximum valueof the gains of the second control object model 12. Hence, the scalingmatrix T can be easily determined. Thereby, the multivariable controllercan be easily designed in accordance with the H ∞ logic which issuperior in the set value followup characteristic and disturbancesuppression, and capable of stabilizing even when the control object isvaried or the control object model has some error.

THIRD EMBODIMENT

In the first and second embodiments, the magnitudes of the gains of themodel are made consistent by the scaling matrix T, so that the weightsof control for the controlled variables are even. However, in practice,there is the problem that the controlled variables y interfere with eachother in the control, and the control is unstable, whereby it may besometimes required to coordinate the weights of control for thecontrolled variables. Thus, in this second embodiment, a weight matrix Sis introduced to weight the controlled variables y directly.

FIG. 11 is a block diagram showing the configuration of a design deviceof a controller in a third embodiment of the present invention. FIG. 12is a block diagram showing the configuration of the generalized plant inthis third embodiment. The design device of FIG. 11 is the design deviceof the first or second embodiment as shown in FIG. 1, comprisingadditionally a control variable weight input unit 7 for inputting theweight for the controlled variable y, a control variable weightregistration unit 8 for registering the control variable weight in thedevice, and a weight matrix calculation unit 9 for calculating theweight matrix S based on the control variable weight. Also, thegeneralized plant of FIG. 12 is the generalized plant of the first orsecond embodiment as shown in FIG. 4, with a block (control variableweight adjusting means) 20 representing the weight matrix S. In thisembodiment, the control variable weight adjusting means 20 (weightmatrix S) is provided inside a closed loop system composed of themanipulated variable weight adjusting means 16 (scaling matrix T), thecontrol object model 11 and the controller K. A general expression ofthe weight matrix S is shown in the following expression.

$\begin{matrix}{S = \begin{bmatrix}S_{1} & 0 & 0 & \cdots & 0 \\0 & S_{2} & 0 & \cdots & 0 \\0 & 0 & S_{3} & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \vdots \\0 & 0 & 0 & \cdots & S_{L}\end{bmatrix}} & (29)\end{matrix}$

Assuming that the number of controlled variables y is L (L is a positiveinteger), the weight matrix S is L×L matrix. Element S_(L) Of the weightmatrix S is the weight for the L-th controlled variable y_(L). Eachelement S_(L) is determined in the following expression.

$\begin{matrix}{S_{L} = \frac{W_{y\; L}}{\max\left( {W_{y\; 1},W_{y\; 2},\cdots,W_{y\; L}} \right)}} & (30)\end{matrix}$

In the expression (30), W_(yL) is the control variable weight for theL-th controlled variable y_(L). In this way, each controlled variablecan be directly weighted by the weight matrix S.

Next, in this third embodiment, the configuration of the generalizedplant is modified as shown in FIG. 12, so that the following expressioncan hold.e₁=Se  (31)

From the expressions (31) and (3), the expressions (6) and (7) can berewritten in the following expression.

$\begin{matrix}\begin{matrix}{{\overset{.}{x}}_{\alpha} = {{A_{\alpha}x_{\alpha}} + {B_{\alpha}{\mathbb{e}}_{1}}}} \\{= {{A_{\alpha}x_{\alpha}} + {B_{\alpha}S\; C_{p}x_{p}} + {B_{\alpha}{SD}_{p\; 1}{Mw}} + {B_{\alpha}{SD}_{p\; 2}{Tu}} - {B_{\alpha}{Sr}}}}\end{matrix} & (32)\end{matrix}$

$\begin{matrix}\begin{matrix}{{\mathbb{e}}_{2} = {{C_{\alpha}x_{\alpha}} + {D_{\alpha}{\mathbb{e}}}}} \\{= {{C_{\alpha}x_{\alpha}} + {D_{\alpha}S\; C_{p}x_{p}} + {D_{\alpha}{SD}_{p\; 1}{Mw}} + {D_{\alpha}{SD}_{p\; 2}{Tu}} - {D_{\alpha}{Sr}}}}\end{matrix} & (33)\end{matrix}$

Thereby, the expressions (8), (9) and (10) can be rewritten in thefollowing expression.

$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}t}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} = {{\begin{bmatrix}A_{p} & 0 \\{B_{\alpha}S\; C_{p}} & A_{\alpha}\end{bmatrix}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} + {\begin{bmatrix}0 & {B_{p\; 1}M} & {B_{p\; 2}T} \\{{- B_{\alpha}}S} & {B_{\alpha}{SD}_{p\; 1}M} & {B_{\alpha}{SD}_{p\; 2}T}\end{bmatrix}\begin{bmatrix}r \\w \\u\end{bmatrix}}}} & (34)\end{matrix}$

$\begin{matrix}{\begin{bmatrix}{z_{1}}^{\prime} \\{z_{2}}^{\prime}\end{bmatrix} = {{\begin{bmatrix}{D_{\alpha}S\; C_{p}} & C_{\alpha} \\0 & 0\end{bmatrix}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} + {\begin{bmatrix}{{- D_{\alpha}}S} & {D_{\alpha}{SD}_{p\; 1}M} & {D_{\alpha}{SD}_{p\; 2}T} \\0 & 0 & I\end{bmatrix}\begin{bmatrix}r \\w \\u\end{bmatrix}}}} & (35)\end{matrix}$

$\begin{matrix}{{\mathbb{e}}_{2} = {{\begin{matrix}\left\lbrack {D_{\alpha}S\; C_{p}} \right. & \left. C_{\alpha} \right\rbrack\end{matrix}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} + {\begin{bmatrix}{{- D_{\alpha}}S} & {D_{\alpha}{SD}_{p\; 1}M} & {D_{\alpha}{SD}_{p\; 2}T}\end{bmatrix}\begin{bmatrix}r \\w \\u\end{bmatrix}}}} & (36)\end{matrix}$

Representing the expressions (34), (35) and (36) in the Doile's notationas in the expression (11), the parameter A in the expression (11) can berepresented as follows:

$\begin{matrix}{A = \begin{bmatrix}A_{p} & 0 \\{B_{\alpha}S\; C_{p}} & A_{\alpha}\end{bmatrix}} & (37)\end{matrix}$

The parameters B₁ and B₂ can be represented in the following expression.

$\begin{matrix}{{B_{1} = \begin{bmatrix}0 & {B_{p\; 1}M} \\{{- B_{\alpha}}S} & {B_{\alpha}{SD}_{p\; 1}M}\end{bmatrix}},{B_{2} = \begin{bmatrix}{B_{p\; 2}T} \\{B_{\alpha}{SD}_{p\; 2}T}\end{bmatrix}}} & (38)\end{matrix}$

The parameters C₁ and C₂ can be represented in the following expression.

$\begin{matrix}{{C_{1} = \begin{bmatrix}{D_{\alpha}S\; C_{p}} & C_{\alpha} \\0 & 0\end{bmatrix}},{C_{2} = \left\lfloor \begin{matrix}{D_{\alpha}S\; C_{p}} & \left. C_{\alpha} \right\rfloor\end{matrix} \right.}} & (39)\end{matrix}$

The parameters D₁₁, D₁₂, D₂₁, and D₂₂ can be represented in thefollowing expression.

$\begin{matrix}{{{D_{11} = \begin{bmatrix}{{- D_{\alpha}}S} & {D_{\alpha}{SD}_{p\; 1}M} \\0 & 0\end{bmatrix}},{D_{12} = \begin{bmatrix}{D_{\alpha}{SD}_{p\; 2}T} \\I\end{bmatrix}}}{{D_{21} = \begin{bmatrix}{{- D_{\alpha}}S} & {D_{\alpha}{SD}_{p\; 1}M}\end{bmatrix}},{D_{22} = {D_{\alpha}{SD}_{p\; 2}T}}}} & (40)\end{matrix}$

The sensitivity weight W_(s) and the complementary sensitivity weightW_(t) are designed in the same manner as in the first embodiment, andmultiplied by the output part of the expression (11), and through γiteration, the controller K is obtained in the state spacerepresentation. The actual controller mounted in the plants such asdistillation tower is the controller K multiplied by the weight matrixS, the weight α⁻¹I and the scaling matrix T, as shown in FIG. 13.

Referring now to FIG. 11, the above operation will be described below.The operation of the control object model input unit 1, the controlobject model registration unit 2, the frequency response calculationunit 4 and the scaling matrix calculation unit 5 is exactly the same asin the first or second embodiment. A memory unit 3 a stores the formulaeof the generalized plant of FIG. 12 and the formulae of the controlobject model that is part of the generalized plant, as described in theexpressions (1) to (5), (11), and (32) to (40). The control variableweight W_(yL) for the L-th controlled variable y_(L) is set in thecontrol variable weight input unit 7 by the user of the design device.This control variable weight W_(yL) is set for each controlled variabley. The control variable weight registration unit 8 outputs the controlvariable weight W_(yL) input from the control variable weight input unit7 to the weight matrix calculation unit 9. The weight matrix calculationunit 9 calculates the weight matrix S, based on the control variableweight W_(yL), employing the expressions (29) and (30), and outputs itto a controller calculation unit 6 a. The controller calculation unit 6a registers the scaling matrix T and the weight matrix S in the formulaeof the generalized plant stored in the memory unit 3 a, and through γiteration, calculates the parameters of the controller K. At this time,the complementary sensitivity weight W_(t), the sensitivity weight W_(s)and the scaling matrix M are preset in the generalized plant in thememory unit 3 a. In this way, the controller K can be designed.

As described above, in this third embodiment, each controlled variable ycan be directly weighted by introducing the weight matrix S. Thereby,the controller with higher control performance and enhanced stabilitycan be designed. Also, there is no need of providing the scaling matrixT with a role of weighting the controlled variable y by introducing theweight matrix S.

FOURTH EMBODIMENT

In the third embodiment, the control variable weight adjusting means 20(weight matrix S) is provided inside the closed loop system, but may beprovided outside the closed loop system. FIG. 14 is a block diagramshowing the configuration of the generalized plant in a fourthembodiment of the present invention. In this fourth embodiment, thecontrol variable weight adjusting means 20 (weight matrix S) is providedat the former stage of the frequency sensitivity weight adjusting means13. A method for determining the weight matrix S is exactly the same asthe method for determining in the third embodiment as described in theexpressions (29) and (30).

Next, in this fourth embodiment, the configuration of the generalizedplant is modified as shown in FIG. 14, so that the expression (4) can berewritten into the following expression.z₁′=Se₂  (41)

Thereby, the expression (9) can be rewritten in the followingexpression.

$\begin{matrix}{\begin{bmatrix}{z_{1}}^{\prime} \\{z_{2}}^{\prime}\end{bmatrix} = {{\begin{bmatrix}{{SD}_{\alpha}C_{p}} & {SC}_{\alpha} \\0 & 0\end{bmatrix}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} + {\begin{bmatrix}{- {SD}_{\alpha}} & {{SD}_{\alpha}D_{p\; 1}M} & {{SD}_{\alpha}D_{p\; 2}T} \\0 & 0 & I\end{bmatrix}\begin{bmatrix}r \\w \\u\end{bmatrix}}}} & (42)\end{matrix}$

Representing the expressions (8), (10) and (42) in the Doile's notationas in the expression (11), the parameters C₁ and C₂ in the expression(11) can be represented in the following expression.

$\begin{matrix}{{C_{1} = \begin{bmatrix}{{SD}_{\alpha}C_{p}} & {SC}_{\alpha} \\0 & 0\end{bmatrix}},{C_{2} = \left\lfloor \begin{matrix}{D_{\alpha}C_{p}} & \left. C_{\alpha} \right\rfloor\end{matrix} \right.}} & (43)\end{matrix}$

Also, the parameters D₁₁, D₁₂, D₂₁ and D₂₂ can be represented in thefollowing expression.

$\begin{matrix}{{{D_{11} = \begin{bmatrix}{{- S}\; D_{\alpha}} & {S\; D_{\alpha}D_{p\; 1}M} \\0 & 0\end{bmatrix}},{D_{12} = \begin{bmatrix}{S\; D_{\alpha}D_{p\; 2}T} \\I\end{bmatrix}}}{{D_{21} = \begin{matrix}\left\lfloor {- D_{\alpha}} \right. & \left. {D_{\alpha}D_{p\; 1}M} \right\rfloor\end{matrix}},{D_{22} = {D_{\alpha}D_{p\; 2}T}}}} & (44)\end{matrix}$

The parameters A, B₁ and B₂ are the same as shown in the expressions(12) and (13). In this fourth embodiment, the actual controller mountedin the plants such as distillation tower is the controller K multipliedby the weight α⁻¹ and the scaling matrix T, as shown in FIG. 6.

In this fourth embodiment, the configuration of the design device isalmost the same as in the third embodiment. Thus, the operation of thedesign device of this fourth embodiment will be described below withreference to FIG. 11.

The operation of the control object model input unit 1, the controlobject model registration unit 2, the frequency response calculationunit 4 and the scaling matrix calculation unit 5 is exactly the same asin the first or second embodiment. A memory unit 3 a stores the formulaeof the generalized plant of FIG. 14 and the formulae of the controlobject model that is part of the generalized plant, as described in theexpressions (1) to (3), (5) to (8), (10) to (13), and (41) to (44). Theoperation of the control variable weight input unit 7, the controlvariable weight registration unit 8 and the weight matrix calculationunit 9 is exactly the same as in the third embodiment. The controllercalculation unit 6 a registers the scaling matrix T and the weightmatrix S in the formulae of the generalized plant stored in the memoryunit 3 a, and through γ iteration, calculates the parameters of thecontroller K. In this way, the controller K can be designed.

In this fourth embodiment, the control variable weight adjusting means20 (weight matrix S) is provided at the former stage of the frequencysensitivity weight adjusting means 13, but may be provided at the latterstage of the frequency sensitivity weight adjusting means 13. Since themagnitudes of the gains of the control object model 11 are madeconsistent by the scaling matrix T, the weight matrix S can be easilyadjusted, and it is prerequisite that the method for determining thescaling matrix T as described in the first or second embodiment isemployed in the third or fourth embodiment.

FIFTH EMBODIMENT

FIG. 15 is a block diagram showing the configuration of a design deviceof a controller in a fifth embodiment of the present invention. Thedesign device of FIG. 15 comprises a transient response parameter inputunit 101 for inputting the transient response parameters indicating thetransient response characteristics of a closed loop system composed ofthe control objects and the controller, a transient response parameterregistration unit 102 for registering the transient response parameterswithin the device, a closed loop transfer function calculation unit 103for calculating the transient response characteristics of the closedloop system based on the transient response parameters input from thetransient response parameter registration unit 102, a frequencysensitivity weight calculation unit 104 for calculating the frequencysensitivity weight for determining the set value followup characteristicof the closed loop system, based on the transient responsecharacteristics of the closed loop system, and a controller calculationunit 105 for calculating the parameters of the controller by applyingthe frequency sensitivity weight to the preset generalized plants. Thetransient response parameter input unit 101, the transient responseparameter registration unit 102 and the closed loop transfer functioncalculation unit 103 constitute setting means for setting the transientresponse characteristics of the closed loop system.

An algorithm for designing the controller in accordance with the H ∞logic is designed based on the generalized plants as represented usingthe control objects. Therefore, the generalized plants will be firstdescribed. FIG. 16 is a block diagram showing the configuration of amodel of an actual control object as represented in the formulae. Anumerical model of the control object as shown in FIG. 16 is composed ofa model 111 for the manipulated variable u and a model 112 for thedisturbance w. Reference sign Pu denotes a transfer function of themodel 111 and reference sign Pw denotes a transfer function of the model112. The models 111, 112 are obtained as a result of the modelidentification using the data obtained by a step response test for theactual control object. The controlled variable y that is the output ofcontrol object is a sum of the outputs from the models 111 and 112.

FIG. 17 shows the configuration of the conventional generalized plantscontaining such numerical model of control object. The generalizedplants involve providing a frequency weight called a sensitivity weightW_(s) used to determine the set value followup characteristic and afrequency weight called a complementary sensitivity weight W_(t) used todetermine the robust stability, and introducing the set value r, and theoutputs z1, z2 in addition to the input (manipulated variable) u ofcontrol object, input (disturbance) w, and output (controlled variable)y, to realize both the set value followup characteristic and the robuststability as shown in FIG. 17. Deviation e (=y−r) is an observationquantity or the input into the controller (not shown). Reference numeral113 denotes a block representing the sensitivity weight W_(s), andreference sign Z₁ denotes an output for effecting the evaluation of theset value followup characteristic. Also, reference numeral 114 denotes ablock representing the complementary sensitivity weight W_(t), andreference sign Z₂ denotes an output for effecting the evaluation of therobust stability.

Conventionally, in the generalized plants as shown in FIG. 17, thecomplementary sensitivity weight W_(t) is determined by estimating theuncertainty of the models on the basis of the numerical models ofcontrol objects, the sensitivity weight W_(s) is determined by directlyspecifying the frequency characteristic in view of the followup abilityto the set value r, whereby the parameters of the controller aredetermined through the γ iteration. However, since employing thegeneralized plants of FIG. 17, the controller is designed on the basisof a larger gain due to differences in the gains for the plant outputsof the manipulated variables, the obtained controller is likely to bevery conservative, or excessively stable. Also, since the set valuefollowup characteristic and the disturbance response characteristic areusually reciprocal, it is preferred to design with the weight accordingto the purpose rather than with the same weight. Further, in thegeneralized plants of FIG. 17 containing no integral element, somesteady-state deviation arises. If the sensitivity weight W_(s) isprovided with the integral characteristic, the controller can have theintegral characteristic, but the generalized plants become unstable, notleading to the normal H ∞ problem.

Therefore, in this fifth embodiment, the generalized plant as shown inFIG. 18 is considered. In FIG. 18, reference sign M denotes a scalingmatrix for adjusting the influence of disturbance won the controlledvariable y, reference sign T denotes a scaling matrix for making themagnitudes of errors of the control objects consistent, and referencesign α⁻¹I denotes a weight for providing the controller with theintegral characteristic to eliminate the steady-state deviation. Herein,α(s) is defined as α(s)=s/(s+a). Where s is a Laplace operator, and a(>0) is any real number. Reference numeral 115 denotes a blockrepresenting the scaling matrix M, reference numeral 116 denotes a blockrepresenting the scaling matrix T, and reference numeral 117 denotes ablock representing the weight α⁻¹I. Deviation e₂ is deviation emultiplied by the weight α⁻¹I, and input into the controller. FIG. 19shows the configuration of a robust control system in which a controllerK is added to the above generalized plant. In FIG. 19, reference numeral118 denotes a block representing the controller K.

The design device of the controller in this fifth embodiment is aimed atdetermining the parameters of the controller K such that the controlledvariable y that is the output of control object follows the set value r,the influence of disturbance w is removed, and the control object isstabilized even if it is fluctuated or there is an error in the model ofcontrol object. The H ∞ control problem can be regarded as the problemof reducing the H ∞ norm (gain) of the transfer function from (r, w) to(z₁, z₂). That is, the set value followup characteristic, the robuststability and the disturbance suppression may be considered in thefollowing way.

(D) Set value followup characteristic: If the H ∞ norm (gain) of thetransfer function from the set value r to deviation e (more correctly,transfer function with the set value r multiplied by the frequencyweight α⁻¹W_(s) from r to z₁) is reduced, the deviation e can bedecreased, so that the set value followup characteristic can be madebetter. Herein, α⁻¹W_(s) is the frequency weight for restricting thefollowup band (e.g., followup only in the low band).

(E) Robust stability: There is an error between the actual controlobject and its model due to characteristic variations of the controlobject or the error at the time of modeling. The maximum value of theerror from the identified model is estimated as Δ(s), and the controllerK is designed such that the H ∞ norm from the set value r to z₂ issmaller than or equal to 1, employing the complementary sensitivityweight W_(t) (s) such that |Δ(jω)|<|W_(t)(jω)| for this error, wherebythe robust stabilization can be achieved.

(F) Disturbance suppression: If the H ∞ norm (gain) of the transferfunction from disturbance w to deviation e (more correctly, transferfunction with the disturbance w multiplied by the frequency weightα⁻¹W_(s) from w to z₁) is reduced, the deviation e can be decreased evenif the disturbance w enters, whereby the disturbance suppression can beameliorated.

Next, it is supposed that the state space representation of thegeneralized plant as shown in FIG. 18 is given by:{dot over (x)}=A _(p) x _(p) +B _(p1) Mw+B _(p2) Tu  (101)y=C _(p) x _(p) +D _(p1) Mw+D _(p2) Tu  (102)

In the above expressions (101) and (102), x_(p) is the quantity ofstate, and A_(p), B_(p1), B_(p2), C_(p), D_(p1) and D_(p2) are theparameters of the numerical models 111, 112 of control objects. From theexpression (102), the deviation e can be obtained in the followingexpression.e=y−r=C _(p) x _(p) +D _(p1) Mw+D _(p2) Tu−r  (103)

With the configuration of the generalized plant as shown in FIG. 18, theoutputs z₁′ and z₂′ can be defined in the following expressions.z₁′=e₂  (104)z₂′=u  (105)

The frequency weight for providing the controller K with the integralcharacteristic can be defined in the following expression, using theexpression (103).

$\begin{matrix}\begin{matrix}{{\overset{.}{x}}_{\alpha} = {{A_{\alpha}x_{\alpha}} + {B_{\alpha}{\mathbb{e}}}}} \\{= {{A_{\alpha}x_{\alpha}} + {B_{\alpha}C_{p}x_{p}} + {B_{\alpha}D_{p\; 1}{Mw}} + {B_{\alpha}D_{p\; 2}{Tu}} - {B_{\alpha}r}}}\end{matrix} & (106)\end{matrix}$

$\begin{matrix}\begin{matrix}{{\mathbb{e}}_{2} = {{C_{\alpha}x_{\alpha}} + {D_{\alpha}{\mathbb{e}}}}} \\{= {{C_{\alpha}x_{\alpha}} + {D_{\alpha}C_{p}x_{p}} + {D_{\alpha}D_{p\; 1}{Mw}} + {D_{\alpha}D_{p\; 2}{Tu}} - {D_{\alpha}r}}}\end{matrix} & (107)\end{matrix}$

In the expressions (106) and (107), x_(α) is the quantity of state ofα⁻¹I, and A_(α), B_(α), C_(α) and D_(α) are the parameters of α⁻¹I.Arranging the above expressions and representing them in a state space,the following three expressions can be obtained.

$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}t}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} = {{\begin{bmatrix}A_{p} & 0 \\{B_{\alpha}C_{p}} & A_{\alpha}\end{bmatrix}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} + {\begin{bmatrix}0 & {B_{p\; 1}M} & {B_{p\; 2}T} \\{- B_{\alpha}} & {B_{\alpha}D_{p\; 1}M} & {B_{\alpha}D_{p\; 2}T}\end{bmatrix}\begin{bmatrix}r \\w \\u\end{bmatrix}}}} & (108)\end{matrix}$

$\begin{matrix}{\begin{bmatrix}z_{1}^{\prime} \\z_{2}^{\prime}\end{bmatrix} = {{\begin{bmatrix}{D_{\alpha}C_{p}} & C_{\alpha} \\0 & 0\end{bmatrix}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} + {\begin{bmatrix}{- D_{\alpha}} & {D_{\alpha}D_{p\; 1}M} & {D_{\alpha}D_{p\; 2}T} \\0 & 0 & I\end{bmatrix}\begin{bmatrix}r \\w \\u\end{bmatrix}}}} & (109)\end{matrix}$

$\begin{matrix}{{\mathbb{e}}_{2} = {{\begin{matrix}\left\lbrack {D_{\alpha}C_{p}} \right. & \left. C_{\alpha} \right\rbrack\end{matrix}\begin{bmatrix}x_{p} \\x_{\alpha}\end{bmatrix}} + {\begin{bmatrix}{- D_{\alpha}} & {D_{\alpha}D_{p\; 1}M} & {D_{\alpha}D_{p\; 2}T}\end{bmatrix}\begin{bmatrix}r \\w \\u\end{bmatrix}}}} & (110)\end{matrix}$

Representing the expressions (108), (109) and (110) in Doile's notation,the following expression can be obtained.

$\begin{matrix}{{G(s)} = \begin{bmatrix}A & B_{1} & B_{2} \\C_{1} & D_{11} & D_{12} \\C_{2} & D_{21} & D_{22}\end{bmatrix}} & (111)\end{matrix}$Parameter A can be represented as follows:

$\begin{matrix}{A = \begin{bmatrix}A_{p} & 0 \\{B_{\alpha}C_{p}} & A_{\alpha}\end{bmatrix}} & (112)\end{matrix}$

Parameters B₁ and B₂ can be represented as follows:

$\begin{matrix}{{B_{1} = \begin{bmatrix}0 & {B_{p\; 1}M} \\{- B_{\alpha}} & {B_{\alpha}D_{p\; 1}M}\end{bmatrix}},{B_{2} = \begin{bmatrix}{B_{p\; 2}T} \\{B_{\alpha}D_{p\; 2}T}\end{bmatrix}}} & (113)\end{matrix}$

Also, parameters C₁ and C₂ can be represented as follows:

$\begin{matrix}{{C_{1} = \begin{bmatrix}{D_{\alpha}C_{p}} & C_{\alpha} \\0 & 0\end{bmatrix}},{C_{2} = \left\lfloor \begin{matrix}{D_{\alpha}C_{p}} & C_{\alpha}\end{matrix} \right\rfloor}} & (114)\end{matrix}$

And the parameters D₁₁, D₁₂, D₂₁, D₂₂ can be represented as follows:

$\begin{matrix}{{{D_{11} = \begin{bmatrix}{- D_{\alpha}} & {D_{\alpha}D_{p\; 1}M} \\0 & 0\end{bmatrix}},{D_{12} = \begin{bmatrix}{D_{\alpha}D_{p\; 2}T} \\I\end{bmatrix}}}{{D_{21} = \begin{matrix}\left\lfloor {- D_{\alpha}} \right. & \left. {D_{\alpha}D_{p\; 1}M} \right\rfloor\end{matrix}},{D_{22} = {D_{\alpha}D_{p\; 2}T}}}} & (115)\end{matrix}$

The sensitivity weight W_(s) and the complementary sensitivity weightW_(t) are designed, and multiplied by the output parts of the expression(111). Through γ iteration, the controller K is obtained in the statespace representation. Herein, the output parts of the expression (111)signify the parts corresponding to the outputs z₁′, z₂′ in FIG. 18.Hence, an output equation of the parameters C₁, D₁₁, D₁₂ of theexpression (111) may be multiplied by a diagonal matrix Q as representedin the following expression having the diagonal elements of sensitivityweight W_(s) and complementary sensitivity weight W_(t) from the leftside. Thus, the parameters of the controller K can be calculated.

$\begin{matrix}{Q = \begin{bmatrix}W_{s} & 0 \\0 & W_{t}\end{bmatrix}} & (116)\end{matrix}$

The controller K is a solution of the H ∞ control problem with thegeneralized plants, and the actual controller mounted on the plants suchas a distillation tower is the controller K multiplied by weight α⁻¹Iand scaling matrix T, as shown in FIG. 20.

As previously described, the H ∞ logic is a design method in thefrequency domain. Therefore, the design is easy in the control of themechanical system, but is hard in the control system untreatable in thefrequency domain such as the process control, and it is difficult tosuitably select the complementary sensitivity weight W_(t) and thesensitivity weight W_(s). In the following, a method for determining thecomplementary sensitivity weight W_(t) in this embodiment will bedescribed below. The control object is varied in the characteristics,depending on the driving conditions. Normally, the control design ismade based on a certain model, but in the robust control design, thevariation of control object and the magnitude of error of modeling arecontained beforehand in the control design, and the controller isdesigned to be stable without much deterioration in the controlperformance even if there is any variation or error. FIG. 21 shows anadditive error for the model 111 of control object. In FIG. 21,reference numeral 119 denotes a block representing the additive error Δ.In the robust control design, a variation in the characteristic of thecontrol object owing to the driving conditions and a model error due tolower dimension of the model 111 are represented as the additive error Δas shown in FIG. 21. If the characteristics of the control object aredeviated from the model 111 due to this additive error Δ, the controlleris designed such that the controller output may be stable. To this end,the complementary sensitivity weight W_(t) may be determined to coverthe additive error Δ. The general expression of this complementarysensitivity weight W_(t) is shown in the following expression. Since thechange of the model 112 is not related with the stability of the system,it is supposed that the model 111 alone is varied in designing thecontroller.

$\begin{matrix}{W_{t} = \begin{bmatrix}W_{t\; 1} & 0 & 0 & \cdots & 0 \\0 & W_{t\; 2} & 0 & \cdots & 0 \\0 & 0 & W_{t\; 3} & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \vdots \\0 & 0 & 0 & \cdots & W_{tN}\end{bmatrix}} & (117)\end{matrix}$

In this fifth embodiment, for the additive error Δ with scaledmagnitude, employing the scaling matrix T, the element of thecomplementary sensitivity weight W_(t) is the maximum value Gmax of thegains of error Δ multiplied by a safety factor δ(δ is equal to 1, forexample). Namely, the elements (weights) W_(t1), W_(t2), W_(t3), . . . ,W_(tN) are defined as follows:W _(t1) =W _(t2) =W _(t3) =W _(tN)=(1+δ)G max  (118)

This embodiment is involved with the multivariable control system, andassuming that the number of manipulated variables u is N (N is apositive integer), the complementary sensitivity weight W_(t) is N×Nmatrix. Where W_(tN) is the weight for the N-th manipulated variableu_(N).

Next, the methods for determining the scaling matrices T, M will bedescribed below. A general expression of the scaling matrix T is shownin the following expression.

$\begin{matrix}{T = \begin{bmatrix}T_{1} & 0 & 0 & \cdots & 0 \\0 & T_{2} & 0 & \cdots & 0 \\0 & 0 & T_{3} & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \vdots \\0 & 0 & 0 & \cdots & T_{N}\end{bmatrix}} & (119)\end{matrix}$

Assuming that the number of manipulated variables u is N, the scalingmatrix T is N×N matrix. Element T_(N) of the scaling matrix T is theweight for the N-th manipulated variable u_(N). Each element T_(N) isdetermined such that the magnitudes of the components of the additiveerror Δ are as equal as possible.

Next, a general expression of the scaling matrix M is shown in thefollowing expression.

$\begin{matrix}{M = \begin{bmatrix}M_{1} & 0 & 0 & \cdots & 0 \\0 & M_{2} & 0 & \cdots & 0 \\0 & 0 & M_{3} & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \vdots \\0 & 0 & 0 & \cdots & M_{J}\end{bmatrix}} & (120)\end{matrix}$

Assuming that the number of disturbance w is J (J is a positiveinteger), the scaling matrix M is J×J matrix. Element M_(J) of thescaling matrix M is the weight for the J-th disturbance w_(J). Eachelement M_(J) is an adjustment parameter for determining the disturbancesuppression performance by adjusting the influence of each disturbancew_(J) on the controlled variable y.

Next, a method for determining the sensitivity weight W_(s) in thisfifth embodiment will be described below. First of all, a generalexpression of the sensitivity weight W_(s) is shown in the followingexpression.

$\begin{matrix}{W_{s} = \begin{bmatrix}W_{s\; 1} & 0 & 0 & \cdots & 0 \\0 & W_{s\; 2} & 0 & \cdots & 0 \\0 & 0 & W_{s\; 3} & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \vdots \\0 & 0 & 0 & \cdots & W_{sL}\end{bmatrix}} & (121)\end{matrix}$

Assuming that the number of controlled variables y is L (L is a positiveinteger), the sensitivity weight W_(s) is L×L matrix. Element W_(sL) ofthe sensitivity weight W_(s) is the weight for the L-th controlledvariable y_(L). In order to determine the sensitivity weight W_(s), aclosed loop system of FIG. 22 in which the robust control system of FIG.19 is simplified is considered. In FIG. 22, reference numeral 111 adenotes a block representing a numerical model P of control object, andreference numeral 113 a denotes a block representing the frequencyweight W_(s)′.

Supposing that the sensitivity function indicating the controlperformance mainly regarding the quick-response property such as the setvalue followup ability or disturbance suppression is S(s), the smallergain |S(jω)| of the sensitivity function S(s) is preferable because themodel variations have less effect on the set value response. If thecontrol specification at each frequency is given by S_(spec) (ω), thefollowing condition concerning the sensitivity function S(s) can beobtained.|S(jω)|<S _(spec)(ω);∀ω  (122)

Where ∀ω means that the expression (122) holds for all frequencies ω.

On one hand, the sensitivity function S(s) corresponds to the transferfunction G_(er) (s) of the closed loop system from the set value r todeviation e, as shown in FIG. 22. Supposing that the transfer functionof control object is P(s) and the transfer function of controller isK(s), the following expression is obtained.

$\begin{matrix}{{S(s)} = {{G_{er}(s)} = \frac{- 1}{1 - {{P(s)}{K(s)}}}}} & (123)\end{matrix}$

Also, the transfer function G_(yr)(s) of the closed loop system of FIG.22 from the set value r to the controlled variable y is obtained in thefollowing expression.

$\begin{matrix}{{G_{yr}(s)} = \frac{{- {P(s)}}{K(s)}}{1 - {{P(s)}{K(s)}}}} & (124)\end{matrix}$

From the expressions (123) and (124), the sensitivity function S(s) canbe obtained in the following expression.S(s)=G _(yr)(s)−1  (125)

In this fifth embodiment, the transient response characteristic of theclosed loop system as shown in FIG. 22, or the transfer functionG_(yr)(s) of the closed loop system from the set value r to thecontrolled variable y is approximated with a first-order lagcharacteristic as in the following expression, giving G_(yr-spec)(s) asthe specification of the transfer function G_(yr) (s)

$\begin{matrix}{{G_{yr\_ spec}\;(s)} = \frac{1}{{T_{sL}s} + 1}} & (126)\end{matrix}$

In the expression (126), T_(sL) is a time constant regarding the L-thcontrolled variable y_(L). Employing G_(yr-spec)(s) in the expression(126), instead of G_(yr)(s) in the expression (125), the followingcontrol specification S_(spec)(s) is obtained.

$\begin{matrix}{{S_{spec}\;(s)} = {- \frac{T_{sL}s}{{T_{sL}s} + 1}}} & (127)\end{matrix}$

In this fifth embodiment, the frequency weight W_(sL)′ (s) regarding theL-th controlled variable y_(L) is set as follows:

$\begin{matrix}{{W_{sL}^{\prime}(s)} = {\frac{1}{S_{spec}\;(s)} = {- \frac{{T_{sL}s} + 1}{T_{sL}s}}}} & (128)\end{matrix}$

The frequency weight W_(sL)′ (s) is W_(sL)(s) multiplied by α⁻¹(s), anddefined as follows:W _(sL)′(s)=α⁻¹(s)W _(sL)(s)  (129)

If the frequency weight W_(sL)′(s) is set as in the expression (128),the expression (122) can be transformed into the following expression.

$\begin{matrix}{{{{S\left( {j\;\omega} \right)}} < \frac{1}{{W_{sL}^{\prime}\left( {j\;\omega} \right)}}};{\forall\omega}} & (130)\end{matrix}$

Further, the expression (130) can be rewritten into the followingexpression, using the H ∞ norm.∥W _(sL)′(s)S(s)∥∞<1  (131)

The expression (131) indicates that the H ∞ norm of the transferfunction of the closed loop system of FIG. 22 from the set value r tothe deviation e (more correctly, the transfer function with the setvalue r multiplied by the frequency weight α⁻¹(s)W_(sL)(s) from r to z₁)is less than 1. This expression (131) is a design index of thecontroller K in view of the set value followup characteristic.Accordingly, by setting the frequency weight W_(sL)′(s) in accordancewith the expression (128), the expression (131) is satisfied, wherebythe controller K can be designed in view of the set value followupcharacteristic. If the expression (128) is transformed, the followingexpression is obtained.

$\begin{matrix}{{W_{sL}^{\prime}(s)} = {\frac{s + a}{s} \cdot \frac{- \left( {{T_{sL}s} + 1} \right)}{T_{sL}\left( {s + a} \right)}}} & (132)\end{matrix}$

The first term on the right side of the expression (132) is α⁻¹(s).Therefore, element W_(sL)(s) of the sensitivity weight W_(s) can becalculated as follows:

$\begin{matrix}{{W_{sL}(s)} = {- \frac{{T_{sL}s} + 1}{T_{sL}\left( {s + a} \right)}}} & (133)\end{matrix}$

Substituting the expression (133), the expression (121) can berepresented as follows:

$\begin{matrix}{W_{s} = \begin{bmatrix}{- \frac{{T_{s\; 1}s} + 1}{T_{s\; 1}\left( {s + a} \right)}} & 0 & 0 & \cdots & 0 \\0 & {- \frac{{T_{s\; 2}s} + 1}{T_{s\; 2}\left( {s + a} \right)}} & 0 & \cdots & 0 \\0 & 0 & {- \frac{{T_{s\; 3}s} + 1}{T_{s\; 3}\left( {s + a} \right)}} & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \vdots \\0 & 0 & 0 & \cdots & {- \frac{{T_{s\; L}s} + 1}{T_{s\; L}\left( {s + a} \right)}}\end{bmatrix}} & (134)\end{matrix}$

In this way, the sensitivity weight W_(s) can be determined. Referringnow to FIG. 15, the above operation will be described below. Thetransient response parameter or the time constant T_(sL) is set in thetransient response parameter input unit 101 by the user of the designdevice. The time constant T_(sL) is set for each controlled variable y.The transient response parameter registration unit 102 outputs the timeconstant T_(sL) input from the transient response parameter input unit101 directly to the closed loop transfer function calculation unit 103.The closed loop transfer function calculation unit 103 calculates thetransfer function G_(yr) (s) of the closed loop system as shown in FIG.22 from the set value r to the controlled variable y by substituting theinput time constant T_(sL) into the expression (126), and outputs it tothe frequency sensitivity weight calculation unit 104. Subsequently, thefrequency sensitivity weight calculation unit 104 calculates thesensitivity weight W_(s), based on the transfer function G_(yr)(s),employing the expressions (125), (127), (128), and (132) to (134), andoutputs it to the controller calculation unit 105. The memory unit 106stores the formulae of generalized plants as described in theexpressions (101) to (115) and shown in FIG. 18. The controllercalculation unit 105 registers the sensitivity weight W_(s) in theformulae of the generalized plants stored in the memory unit 106, andthrough γ iteration, calculates the parameters of the controller K. Atthis time, the complementary sensitivity weight W_(t), and the scalingmatrices T, M are preset in the generalized plants in the memory unit106. In this way, the controller K can be designed.

As previously described, in the process control field, it is difficultto give the control specification in the frequency domain, and todetermine the frequency sensitivity weight W_(s). On the contrary, inthis embodiment, the transient response characteristic (transferfunction G_(yr)(s)) of the closed loop system is set, and the frequencyweight W_(s) can be calculated, based on this transient responsecharacteristic. Thereby, even in the process control field, where thefrequency response characteristic is difficult to give as the controlspecification, the controller can be designed in accordance with the H ∞logic. Consequently, the multivariable control system can be easilydesigned in view of the variations of the control objects and theuncertainty of the numerical models. Also, the controller can berealized utilizing the features of the H ∞ logic with smallercomputational load during execution of the control, and with which thesmall-scale control system can be implemented. By approximating thetransient response characteristic of the closed loop system with afirst-order lag characteristic, the parameters for the control designare intuitively understandable to the designer. Therefore, the designdevice that is easily understood and used by the designer can berealized. Since the parameters are intuitively understandable to thedesigner, the design device that is capable of changing the design canbe realized in the case where the controller is redesigned afterdesigned once.

SIXTH EMBODIMENT

In the fifth embodiment, the transient response characteristic of theclosed loop system is approximated with a first-order lag, and thesensitivity weight W_(s) is determined by specifying the time constantof the closed loop system for the controlled variables. In this case,albeit the control design based on the transient responsecharacteristic, only one design parameter is specified for eachcontrolled variable, and the obtained control system does notnecessarily operate with the first-order lag as specified. Consideringthe practicality of the multivariable control, it is not favorable toincrease the number of design parameters, but if the degree of freedomfor the design is increased by making the meanings of design parametersclear, it may not be necessarily a false policy. Thus, in this sixthembodiment, the controller is designed by approximating thecharacteristics of the closed loop system with a second-order system ascommonly used, and specifying the transient response characteristicparameters.

Firstly, the transient response of the closed loop system of FIG. 22 foreach controlled variable is approximated with the second-order system inthe following expression.

$\begin{matrix}{\frac{y}{r} = \frac{\omega_{n}^{2}}{s^{2} + {2\zeta\;\omega_{n}s} + \omega_{n}^{2}}} & (135)\end{matrix}$

In the expression (135), ζ(ζ>0) is a damping coefficient, and ω_(n)(ζ>0) is a natural frequency. The solution of the characteristicequation of the system as indicated in the expression (135) is obtainedin the following expression.

$\begin{matrix}{s = {{{- \zeta}\;\omega_{n}} \pm {\omega_{n}\sqrt{\zeta^{2} - 1}}}} & (136)\end{matrix}$

Giving a step input (r (s)=1/s), the controlled variable y(s) that is anoutput of the closed loop system can be obtained in the followingexpression.

$\begin{matrix}\begin{matrix}{{y(s)} = {\frac{\omega_{n}^{2}}{\left( {s - s_{1}} \right)\left( {s - s_{2}} \right)} \cdot \frac{1}{s}}} \\{= {\frac{1}{s} + \frac{C_{1}}{s + {\zeta\;\omega_{n}} - {\omega_{n}\sqrt{\zeta^{2} - 1}}} + \frac{C_{2}}{s + {\zeta\;\omega_{n}} + {\omega_{n}\sqrt{\zeta^{2} - 1}}}}}\end{matrix} & (137)\end{matrix}$

The parameters s₁, s₂, C₁, C₂ in the expression (137) are given in thefollowing expression.

$\begin{matrix}{{{s_{1} = {{{- \zeta}\;\omega_{n}} + {\omega_{n}\sqrt{\zeta^{2} - 1}}}},{s_{2} = {{{- \zeta}\;\omega_{n}} - {\omega_{n}\sqrt{\zeta^{2} - 1}}}}}{{C_{1} = \frac{{- \zeta} - \sqrt{\zeta^{2} - 1}}{2\sqrt{\zeta^{2} - 1}}},{C_{2} = \frac{\zeta - \sqrt{\zeta^{2} - 1}}{2\sqrt{\zeta^{2} - 1}}}}} & (138)\end{matrix}$

Consider the most important damped oscillation (0<ζ<1) in thesecond-order system characteristic. Then, the parameters C₁, C₂ aregiven in the following expression.

$\begin{matrix}{{C_{1} = \frac{{\mathbb{i}}\left( {\zeta + {{\mathbb{i}}\sqrt{1 - \zeta^{2}}}} \right)}{2\sqrt{1 - \zeta^{2}}}},{C_{2} = \frac{{\mathbb{i}}\left( {{- \zeta} + {{\mathbb{i}}\sqrt{1 - \zeta^{2}}}} \right)}{2\sqrt{1 - \zeta^{2}}}}} & (139)\end{matrix}$

Thereby, the time response of the closed loop system as shown in FIG. 22can be represented in the following expression.

$\begin{matrix}\begin{matrix}{{y(t)} = {1 + {{\frac{{\mathbb{i}}\left( {\zeta + {{\mathbb{i}}\sqrt{1 - \zeta^{2}}}} \right)}{2\sqrt{1 - \zeta^{2}}} \cdot {\exp\left( {{{- \zeta}\;\omega_{n}} + {{\mathbb{i}}\;\omega_{n}\sqrt{1 - \zeta^{2}}}} \right)}}t} +}} \\{{\frac{{\mathbb{i}}\left( {{- \zeta} + {{\mathbb{i}}\sqrt{1 - \zeta^{2}}}} \right)}{2\sqrt{1 - \zeta^{2}}} \cdot {\exp\left( {{{- \zeta}\;\omega_{n}} - {{\mathbb{i}}\;\omega_{n}\sqrt{1 - \zeta^{2}}}} \right)}}t} \\{= {1 + {\frac{{\mathbb{i}}\left( {\zeta + {{\mathbb{i}}\sqrt{1 - \zeta^{2}}}} \right)}{2\sqrt{1 - \zeta^{2}}} \cdot {{\mathbb{e}}^{{- \zeta}\;\omega_{n}t}\left( {{\cos\;\omega_{n}\sqrt{1 - \zeta^{2}}t} + {{\mathbb{i}}\;\sin\;\omega_{n}\sqrt{1 - \zeta^{2}}t}} \right)}} +}} \\{\frac{{\mathbb{i}}\left( {{- \zeta} + {{\mathbb{i}}\sqrt{1 - \zeta^{2}}}} \right)}{2\sqrt{1 - \zeta^{2}}} \cdot {{\mathbb{e}}^{{- \zeta}\;\omega_{n}t}\left( {{\cos\;\omega_{n}\sqrt{1 - \zeta^{2}}t} - {{\mathbb{i}}\;\sin\;\omega_{n}\sqrt{1 - \zeta^{2}}t}} \right)}} \\{= {1 - {{\mathbb{e}}^{{- \zeta}\;\omega_{n}t}\left( {{\cos\;\omega_{n}\sqrt{1 - \zeta^{2}}t} + {\frac{\zeta}{\sqrt{1 - \zeta^{2}}}\sin\;\omega_{n}\sqrt{1 - \zeta^{2}}t}} \right)}}} \\{{= {1 - {\frac{{\mathbb{e}}^{{- \zeta}\;\omega_{n}t}}{\sqrt{1 - \zeta^{2}}}{\sin\left( {{\omega_{n}\sqrt{1 - \zeta^{2}}t} + \psi} \right)}}}},{\psi = {\cos^{- 1}\zeta}}}\end{matrix} & (140)\end{matrix}$

One embodiment of the time response of the closed loop system asrepresented in the expression (140) is shown in FIG. 23. FIG. 23 showshow the controlled variable y behaves when a 100% step set value r isgiven at time 0. The parameters representing the typical transientresponse characteristic of the second-order system corresponding to thistime response include a rise time t_(r) for which the controlledvariable y reaches the same value as the set value r (herein 100%), theovershoot OS that is an extreme value of the transient deviation firstlytaken after the controlled variable y exceeds the set value r, theovershoot time t_(p) for which the controlled variable y reaches theovershoot OS, the settling time _(q) for which the controlled variable yis contained within 5% of a range of the set value r, and the dampingratio DR that is the ratio of a₁ to a₂ as shown in FIG. 23. The risetime t_(r) can be represented in the following expression, employing thedamping coefficient ζ and the natural frequency ω_(n).

$\begin{matrix}{t_{r} = {\frac{1}{\omega_{n}\sqrt{1 - \zeta^{2}}}\left( {\pi - {\cos^{- 1}\zeta}} \right)}} & (141)\end{matrix}$

Similarly, the overshoot time t_(p) is indicated below.

$\begin{matrix}{t_{p} = \frac{\pi}{\omega_{n}\sqrt{1 - \zeta^{2}}}} & (142)\end{matrix}$

The settling time t_(q) can be obtained in the following expression.

$\begin{matrix}{t_{q} = \frac{3}{\zeta\;\omega_{n}}} & (143)\end{matrix}$

Also, the overshoot OS can be obtained in the following expression,employing the damping coefficient ζ.

$\begin{matrix}{{OS} = {\exp\left( \frac{{- \zeta}\;\pi}{\sqrt{1 - \zeta^{2}}} \right)}} & (144)\end{matrix}$

And the damping ratio DR can be obtained in the following expression.

$\begin{matrix}{{DR} = {{a_{2}/a_{1}} = {\exp\left( {{- 2}\pi\;{\zeta/\sqrt{1 - \zeta^{2}}}} \right)}}} & (145)\end{matrix}$

From the expression (135), the transfer function G_(yr)(s) of the closedloop system of FIG. 22 from the set value r to the controlled variable ycan be obtained in the following expression.

$\begin{matrix}{{G_{yr}(s)} = \frac{\omega_{n}^{2}}{s^{2} + {2\zeta\;\omega_{n}s} + \omega_{n}^{2}}} & (146)\end{matrix}$

If the frequency weight W_(sL)′ (s) is set in the following expression,as in the fifth embodiment, the expression (131) is satisfied, and thecontroller K can be designed in view of the set value followupcharacteristic.

$\begin{matrix}{{W_{sL}^{\prime}(s)} = {\frac{1}{S_{spec}(s)} = {\frac{1}{{G_{yr}(s)} - 1} = \frac{1}{\frac{\omega_{nL}^{2}}{S^{2} + {2\zeta_{L}\omega_{nL}s} + \omega_{nL}^{2}} - 1}}}} & (147)\end{matrix}$

In the expression (147), ζ_(L) is the damping coefficient for the L-thcontrolled variable y_(L), and ω_(nL) is the natural frequency for thecontrolled variable y_(L). The expression (147) can be transformed intothe following expression.

$\begin{matrix}\begin{matrix}{{W_{sL}^{\prime}(s)} = \frac{s^{2} + {2\zeta_{L}\omega_{nL}s} + \omega_{nL}^{2}}{{- s^{2}} - {2\zeta_{L}\omega_{nL}s}}} \\{= {\frac{s + a}{s} \cdot \frac{- \left( {s^{2} + {2\zeta_{L}\omega_{nL}s} + \omega_{nL}^{2}} \right)}{\left( {s + {2\zeta_{L}\omega_{nL}}} \right)\left( {s + a} \right)}}}\end{matrix} & (148)\end{matrix}$

The first term on the right side of the expression (148) is α⁻¹ (s).Accordingly, the element W_(sL) (s) of the sensitivity weight W_(s) canbe calculated in the following expression.

$\begin{matrix}{{W_{sL}(s)} = {- \frac{s^{2} + {2\zeta_{L}\omega_{nL}s} + \omega_{nL}^{2}}{\left( {s + {2\zeta_{L}\omega_{nL}}} \right)\left( {s + a} \right)}}} & (149)\end{matrix}$

Substituting the expression (149), the expression (121) can berepresented as follows:

$\begin{matrix}{W_{s} = \begin{bmatrix}{- \frac{s^{2} + {2\zeta_{1}\omega_{n\; 1}s} + \omega_{n\; 1}^{2}}{\left( {s + {2\zeta_{1}\omega_{n\; 1}}} \right)\left( {s + a} \right)}} & 0 & \cdots & 0 \\0 & {- \frac{s^{2} + {2\zeta_{2}\omega_{n\; 2}s} + \omega_{n\; 2}^{2}}{\left( {s + {2\zeta_{2}\omega_{n\; 2}}} \right)\left( {s + a} \right)}} & \cdots & 0 \\\vdots & \vdots & ⋰ & \vdots \\0 & 0 & \cdots & {- \frac{s^{2} + {2\zeta_{L}\omega_{n\; L}s} + \omega_{n\; L}^{2}}{\left( {s + {2\zeta_{L}\omega_{n\; L}}} \right)\left( {s + a} \right)}}\end{bmatrix}} & (150)\end{matrix}$

To calculate the sensitivity weight W_(s) employing the expression(150), it is required to obtain the damping coefficient ζ_(L) and thenatural frequency ω_(nL). To obtain the damping coefficient ζ_(L) andthe natural frequency ω_(nL), two items are selected from among the risetime t_(r), the overshoot time t_(p), the settling time t_(q), theovershoot OS and the damping ratio DR, and the two selected parametervalues are set. In this sixth embodiment, the method for calculating thedamping coefficient ζ_(L) and the natural frequency ω_(nL), as oneembodiment, will be described with an instance of employing the risetime t_(r) and the overshoot OS. With the overshoot OS as indicated inthe expression (144), the damping coefficient ζ_(L) can be obtained inthe following expression.

$\begin{matrix}{\zeta_{L} = \sqrt{\frac{\left( {\ln\left( {OS}_{L} \right)} \right)^{2}}{\pi^{2} + \left( {\ln\left( {OS}_{L} \right)} \right)^{2}}}} & (151)\end{matrix}$

In the expression (151), OS_(L) is the overshoot for the L-th controlledvariable y_(L). Also, from the rise time t_(r) as indicated in theexpression (141), the natural frequency ω_(nL) can be obtained in thefollowing expression.

$\begin{matrix}{\omega_{nL} = \frac{\pi - {\cos^{- 1}\zeta_{L}}}{t_{rL}\sqrt{1 - \zeta_{L}^{2}}}} & (152)\end{matrix}$

In the expression (152), t_(rL) is the rise time for the L-th controlledvariable y_(L). In this way, if the value of overshoot OS_(L) is set,the damping coefficient ζ_(L) can be calculated from the expression(151), and if the value of the rise time t_(rL) is set, the naturalfrequency ω_(nL) can be calculated from the expression (152). In theabove way, the sensitivity weight W_(s) can be determined.

In this sixth embodiment, the configuration of the design device is mostthe same as in the fifth embodiment. Thus, the operation of the designdevice of this sixth embodiment will be described below with referenceto FIG. 15. The transient response parameters, i.e., the overshootOS_(L) and the rise time t_(rL), are set into the transient responseparameter input unit 101 by the user of the design device. Thisovershoot OS_(L) and the rise time t_(rL) are set for each controlledvariable y. The transient response parameter registration unit 102outputs the overshoot OS_(L) and the rise time t_(rL) input fromtransient response parameter input unit 101 directly to the closed looptransfer function calculation unit 103. The closed loop transferfunction calculation unit 103 calculates the damping coefficient ζ_(L)and the natural frequency ω_(nL), based on the overshoot OS_(L) and therise time t_(rL), employing the expressions (151) and (152).Substituting the damping coefficient ζ_(L) and the natural frequencyω_(nL) into the expression (146), the transfer function G_(yr)(s) iscalculated, and output to the frequency sensitivity weight calculationunit 104. Then, the frequency sensitivity weight calculation unit 104calculates the sensitivity weight W_(s), based on the transfer functionG_(yr)(s), employing the expressions (147) to (150), and outputs it tothe controller calculation unit 105. The operation of the controllercalculation unit 105 and the memory unit 106 is exactly the same as inthe fifth embodiment. In this way, the controller K can be designed.

As described above, in this sixth embodiment, the degree of freedom inthe design can be increased by approximating the transient responsecharacteristic of the closed loop system with the second-order systemcharacteristic, and the controller provided by the design device canhave wider applicability. The first to sixth embodiments are involvedwith the design device which designs the multivariable controller. Also,the design device of the first to sixth embodiments can be implementedon the computer. That is, the computer is equipped with an operationunit, a storage device and an input/output device, and operates as thedesign device in accordance with the program.

INDUSTRIAL APPLICABILITY

As described above, the present invention is suitable for designing themultivariable controller.

1. An apparatus comprising: a design device for designing a controllerin accordance with H infinity (H ∞) control logic, the design deviceemploying generalized plants having control object models formanipulated variables, the device including: storage means for storingsaid generalized plants; parameter calculating means having: settingmeans for setting a transient response characteristic of a closed loopsystem consisting of a control object model and said controller; andfrequency sensitivity weight calculation means for calculating thefrequency sensitivity weight for determining a set value followupcharacteristic of said closed loop system in accordance with thetransient response characteristic of said closed loop system; andcontroller calculation means for deriving said controller by applyingsaid frequency sensitivity weight to said generalized plants stored insaid storage means, said setting means approximates the transientresponse characteristic of said closed loop system with a first-orderlag characteristic.
 2. The design device of controller according toclaim 1, wherein said generalized plants have said control object model,and manipulated variable weight adjusting means for adjusting the inputof manipulated variable into said control object model, which isprovided in the former stage of said control object model, saidparameter calculating means comprises frequency response calculationmeans for calculating the frequency response calculation means forcalculating the frequency responses of said control object models, andscaling matrix calculation means for calculating a scaling matrix T fordetermining the weighting of said manipulated variables with saidmanipulated variable weight adjusting means in accordance with thefrequency responses of said control object models so that the respectivegains of said control object models are consistent, and said controllercalculation means calculates the controller by applying said scalingmatrix T to the manipulated variable weight adjusting means of saidgeneralized plants stored in said storage means.
 3. The design device ofcontroller according to claim 2, wherein said scaling matrix calculationmeans calculates said scaling matrix T as follows: $T = \begin{bmatrix}T_{1} & 0 & 0 & \cdots & 0 \\0 & T_{2} & 0 & \cdots & 0 \\0 & 0 & T_{3} & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \vdots \\0 & 0 & 0 & \cdots & T_{N}\end{bmatrix}$ $\begin{matrix}{T_{N} = {{\frac{1}{L} \cdot \frac{\max\left( {{G_{y\; 1u\; 1}}_{\infty},{G_{y\; 1u\; 2}}_{\infty},\cdots\mspace{14mu},{G_{y\; 1u\; N}}_{\infty}} \right)}{{G_{y\; 1u\; N}}_{\infty}}} +}} \\{{\frac{1}{L} \cdot \frac{\max\left( {{G_{y\; 2u\; 1}}_{\infty},{G_{y\; 2u\; 2}}_{\infty},\cdots\mspace{14mu},{G_{y\; 2u\; N}}_{\infty}} \right)}{{G_{y\; 2u\; N}}_{\infty}}} + \cdots +} \\{\frac{1}{L} \cdot \frac{\max\left( {{G_{y\;{Lu}\; 1}}_{\infty},{G_{y\;{Lu}\; 2}}_{\infty},\cdots\mspace{14mu},{G_{y\;{Lu}\; N}}_{\infty}} \right)}{{G_{y\;{Lu}\; N}}_{\infty}}}\end{matrix}$ where the number of manipulated variables u is N (N is apositive integer), the number of controlled variables y is L (L is apositive integer), and the H ∞ norm of the transfer function of saidcontrol object model from the N-th manipulated variable u_(N) to theL-th controlled variable y_(L) is ∥GyLuN∥∞.
 4. The design device ofcontroller according to claim 1, wherein said generalized plants have afirst control object model for the manipulated variables, a secondcontrol object model for the disturbance, and manipulated variableweight adjusting means for adjusting the input of manipulated variableinto the first control object mode, which is provided in the formerstage of said first control object model, said parameter calculatingmeans comprises frequency response calculation means for calculating thefrequency responses of said first control object model and said secondcontrol object model, and scaling matrix calculation means forcalculation a scaling matrix T for determining the weighting of themanipulated variables with said manipulated variable weight adjustingmeans in accordance with the frequency responses of said first andsecond control object models so that the respective gains of said firstcontrol object model are consistent with the maximum values of the gainsof said second control object model, and said controller calculationmeans calculates the parameters of said controller by applying saidscaling matrix T to the manipulated variable weight adjusting means ofsaid generalized plants tored in said storage means.
 5. The designdevice of controller according to claim 2, wherein the generalizedplants stored in said storage means have control variable weightadjusint means for adjusting the controlled variable weight adjustingmeans for adjusting the controlled variables inside a closed loop systemconsisting of said manipulated variable weight adjusting means, thecontrol object model and the controller, and said design device hassetting means for setting a weight matrix S for determining theweighting of the controlled variables with said control variable weightadjusting means.
 6. The design device of controller according to claim4, wherein the generalized plants stored in said storage means havecontrol variable weight adjusting means for adjusting the controlledvariables inside a closed loop system consisting of said manipulatedvariable weight adjusting means, the first control object model and thecontroller, and said design device has setting means for setting aweight matrix S for determining the weighting of the controlledvariables with said control variable weight adjusting means.
 7. Thedesign device of controller according to claim 2, wherein thegeneralized plants stored in said storage means have control variableweight adjusting means for adjusting the controlled variables in theformer or later stage of frequency sensitivity weight adjusting meansfor determining the set value followup characteristic of a closed loopsystem consisting of said manipulated variable weight adjusting means,the control object model and the controller, and said design device hassetting means for setting a weight matrix S for determining theweighting of the controlled variables with said control variable weightadjusting means.
 8. The design device of controller according to claim4, wherein the generalized plants stored in said storage means havecontrol variable weight adjusting means for adjusting the controlledvariables in the former or latter stage of freqeuncy sensitivity weightadjusting means for determing the set value followup characteristic of aclosed loop system consisting of said manipulated variable weightadjusting means, the first control object model and the controller, andsaid design device has setting means for setting a weight matrix S fordetermining the weighting of the controlled variables with said controlvariable weight adjusting means.
 9. The design device of controlleraccording to claim 1, wherein said frequency sensitivity weightcalculation unit calculates said frequency sensitivity weight inaccordance with the transient response characteristic of said closedloop system, and a design index that the H ∞ norm of the transferfunction of the closed loop system from the set value to the deviationmultiplied by said frequency sensitivity weight is less than
 1. 10. Anapparatus comprising: a design device for designing a controller inaccordance with H infinity (H ∞) control logic, the design deviceemploying generalized plants having control object models formanipulated variables, the device including: storage means for storingsaid generalized plants; parameter calculating means having: settingmeans for setting a transient response characteristic of a closed loopsystem consisting of a control object model and said controller; andfrequency sensitivity weight calculation means for calculating thefrequency sensitivity weight for determining a set value followupcharacteristic of said closed loop system in accordance with thetransient response characteristic of said closed loop system; andcontroller calculation means for deriving said controller by applyingsaid frequency sensitivity weight to said generalized plants stored insaid storage means, wherein said setting means approximates thetransient response characteristic of said closed loop system with asecond-order system characteristic.
 11. The design device of controlleraccording to claim 4, wherein said scaling matrix calculation meanscalculates said scaling matrix T as follows: $T = \begin{bmatrix}T_{1} & 0 & 0 & \cdots & 0 \\0 & T_{2} & 0 & \cdots & 0 \\0 & 0 & T_{3} & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \vdots \\0 & 0 & 0 & \cdots & T_{N}\end{bmatrix}$ $\begin{matrix}{T_{N} = {{\frac{1}{L} \cdot \frac{\max\left( {{G_{y\; 1w\; 1}}_{\infty},{G_{y\; 1w\; 2}}_{\infty},\cdots\mspace{14mu},{G_{y\; 1{wJ}}}_{\infty}} \right)}{{G_{y\; 1u\; N}}_{\infty}}} +}} \\{{\frac{1}{L} \cdot \frac{\max\left( {{G_{y\; 2w\; 1}}_{\infty},{G_{y\; 2w\; 2}}_{\infty},\cdots\mspace{14mu},{G_{y\; 2{wJ}}}_{\infty}} \right)}{{G_{y\; 2u\; N}}_{\infty}}} + \cdots +} \\{\frac{1}{L} \cdot \frac{\max\left( {{G_{y\;{Lw}\; 1}}_{\infty},{G_{y\;{Lw}\; 2}}_{\infty},\cdots\mspace{14mu},{G_{y\;{LwJ}}}_{\infty}} \right)}{{G_{y\;{Lu}\; N}}_{\infty}}}\end{matrix}$ where the number of manipulated variables u is N (N is apositive integer), the number of disturbance w is J (is a positiveinteger), the number of controlled variable y is L (L is a positiveinteger), the H ∞ norm of the transfer function of said first controlobject model from the N-th manipulated variable u_(N) to the L-thcontrolled variable y_(L) is ∥GyLuN∥∞.